{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":8048,"title":"Stress-Strain Properties - 1","description":"This is the first in a series of problems regarding mechanics of materials, in particular, material properties drawn from stress-strain responses. A simplified typical stress-strain response is illustrated below (from quora.com):\r\n\r\nThe yield stress is the pressure required to start deformation of the material being tested. The yield point is the point along the response indicated by the yield stress (vertical axis) and the yield strain (horizontal axis). The response of the material up to this point is elastic, meaning that all deformation is reversible. The elastic modulus (E, also known as modulus of elasticity or Young's modulus) is the slope of this line. Write a function to calculate the elastic modulus for a material, provided the elastic strain and yield stress (yield point).\r\nNext problem: 2 - resilience.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 541px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 332px 270.5px; transform-origin: 332px 270.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 31.5px; text-align: left; transform-origin: 309px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThis is the first in a series of problems regarding mechanics of materials, in particular, material properties drawn from stress-strain responses. A simplified typical stress-strain response is illustrated below (from\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e quora.com):\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 304px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 152px; text-align: center; transform-origin: 309px 152px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline\" src=\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 126px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 63px; text-align: left; transform-origin: 309px 63px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe yield stress is the pressure required to start deformation of the material being tested. The yield point is the point along the response indicated by the yield stress (vertical axis) and the yield strain (horizontal axis). The response of the material up to this point is elastic, meaning that all deformation is reversible. The elastic modulus (E, also known as modulus of elasticity or Young's modulus) is the slope of this line. Write a function to calculate the elastic modulus for a material, provided the elastic strain and yield stress (yield point).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 10.5px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eNext problem: 2 -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/8049-stress-strain-properties-2\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eresilience\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [E] = stress_strain1(S_y,e_y)\r\n\r\nE = 1;\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile \r\n% in a single direction. Some materials, such as metals are generally\r\n% isotropic, whereas others, like composite are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26;\r\nG = 79.3e9; %Pa\r\nE = 200e9; %Pa\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 0.463; %strain-hardening coefficient\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 0.974; %strain-hardening coefficient\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1.845; %strain-hardening coefficient\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n\r\n%% aluminum alloy (6061-T6)%^\u0026\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 0.325; %strain-hardening coefficient\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 0.304; %strain-hardening coefficient 530MPa\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1.870; %strain-hardening coefficient\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.1e9; %Pa\r\ndensity = 1.14; %g/cm^3\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":26769,"edited_by":26769,"edited_at":"2024-03-27T17:40:39.000Z","deleted_by":null,"deleted_at":null,"solvers_count":324,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-30T18:09:31.000Z","updated_at":"2026-03-31T10:50:52.000Z","published_at":"2015-03-30T18:09:58.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis is the first in a series of problems regarding mechanics of materials, in particular, material properties drawn from stress-strain responses. A simplified typical stress-strain response is illustrated below (from\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e quora.com):\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe yield stress is the pressure required to start deformation of the material being tested. The yield point is the point along the response indicated by the yield stress (vertical axis) and the yield strain (horizontal axis). The response of the material up to this point is elastic, meaning that all deformation is reversible. The elastic modulus (E, also known as modulus of elasticity or Young's modulus) is the slope of this line. Write a function to calculate the elastic modulus for a material, provided the elastic strain and yield stress (yield point).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNext problem: 2 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/8049-stress-strain-properties-2\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eresilience\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"target\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationshipId\":\"rId1\"}]},{\"partUri\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"contentType\":\"image/net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"content\":\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":8051,"title":"Stress-Strain Properties - 4","description":"A common measure of the ability of a material to carry load per unit mass is termed strength-to-weight ratio and is calculated by dividing the ultimate tensile strength of the material by its density. This property is key in weight-critical applications, such as aerospace, where many materials with high strength-to-weight ratios are used (e.g., Ni-based superalloys, Ti-based alloys, Al-based alloys, and composites).\r\n\r\nWrite a function to calculate the strength-to-weight ratio for a given material provided its ultimate tensile strength and density.\r\n\r\nPrevious problem: 3 - \u003chttp://www.mathworks.com/matlabcentral/cody/problems/8050-stress-strain-properties-3 qualitative measure of brittleness\u003e. Next problem: 5 - \u003chttp://www.mathworks.com/matlabcentral/cody/problems/8052-stress-strain-properties-5 stiffness-to-weight ratio\u003e.","description_html":"\u003cp\u003eA common measure of the ability of a material to carry load per unit mass is termed strength-to-weight ratio and is calculated by dividing the ultimate tensile strength of the material by its density. This property is key in weight-critical applications, such as aerospace, where many materials with high strength-to-weight ratios are used (e.g., Ni-based superalloys, Ti-based alloys, Al-based alloys, and composites).\u003c/p\u003e\u003cp\u003eWrite a function to calculate the strength-to-weight ratio for a given material provided its ultimate tensile strength and density.\u003c/p\u003e\u003cp\u003ePrevious problem: 3 - \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/8050-stress-strain-properties-3\"\u003equalitative measure of brittleness\u003c/a\u003e. Next problem: 5 - \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/8052-stress-strain-properties-5\"\u003estiffness-to-weight ratio\u003c/a\u003e.\u003c/p\u003e","function_template":"function [StWR] = stress_strain4(S_u,density)\r\n\r\nStWR = 1;\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile\r\n% in a single direction. Some materials, such as metals are generally\r\n% isotropic, whereas others, like composite are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26;\r\nG = 79.3e9; %Pa\r\nE = 200e9; %Pa\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 0.463; %strain-hardening coefficient\r\nStWR_corr = 5.096e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 0.974; %strain-hardening coefficient\r\nStWR_corr = 19.96e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1.845; %strain-hardening coefficient\r\nStWR_corr = 17.18e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%% aluminum alloy (6061-T6)%^\u0026\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 0.325; %strain-hardening coefficient\r\nStWR_corr = 11.11e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 0.304; %strain-hardening coefficient\r\nStWR_corr = 2.466e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1.870; %strain-hardening coefficient\r\nStWR_corr = 5.376e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.1e-2; %Pa\r\ndensity = 1.14; %g/cm^3\r\nStWR_corr = 7.193e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nStWR_corr = 15.23e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nStWR_corr = 34.19e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_u = 400e6; %Pa\r\n\t\tdensity = 7.85; %g/cm^3\r\n\t\tStWR_corr = 5.096e7;\r\n\tcase 2\r\n\t\tS_u = 230e6; %Pa\r\n\t\tdensity = 1.51; %g/cm^3\r\n\t\tStWR_corr = 15.23e7;\r\n\tcase 3\r\n\t\tS_u = 1130e6; %Pa\r\n\t\tdensity = 21.02; %g/cm^3\r\n\t\tStWR_corr = 5.376e7;\r\n\tcase 4\r\n\t\tS_u = 1200e6; %Pa\r\n\t\tdensity = 3.51; %g/cm^3\r\n\t\tStWR_corr = 34.19e7;\r\nend\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_u = 300e6; %Pa\r\n\t\tdensity = 2.7; %g/cm^3\r\n\t\tStWR_corr = 11.11e7;\r\n\tcase 2\r\n\t\tS_u = 900e6; %Pa\r\n\t\tdensity = 4.51; %g/cm^3\r\n\t\tStWR_corr = 19.96e7;\r\n\tcase 3\r\n\t\tS_u = 220e6; %Pa\r\n\t\tdensity = 8.92; %g/cm^3\r\n\t\tStWR_corr = 2.466e7;\r\n\tcase 4\r\n\t\tS_u = 230e6; %Pa\r\n\t\tdensity = 1.51; %g/cm^3\r\n\t\tStWR_corr = 15.23e7;\r\nend\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_u = 300e6; %Pa\r\n\t\tdensity = 2.7; %g/cm^3\r\n\t\tStWR_corr = 11.11e7;\r\n\tcase 2\r\n\t\tS_u = 1200e6; %Pa\r\n\t\tdensity = 3.51; %g/cm^3\r\n\t\tStWR_corr = 34.19e7;\r\n\tcase 3\r\n\t\tS_u = 82e6; %Pa\r\n\t\tdensity = 1.14; %g/cm^3\r\n\t\tStWR_corr = 7.193e7;\r\n\tcase 4\r\n\t\tS_u = 900e6; %Pa\r\n\t\tdensity = 4.51; %g/cm^3\r\n\t\tStWR_corr = 19.96e7;\r\nend\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n","published":true,"deleted":false,"likes_count":4,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":222,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-30T19:24:41.000Z","updated_at":"2026-03-10T20:20:32.000Z","published_at":"2015-03-30T19:24:41.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA common measure of the ability of a material to carry load per unit mass is termed strength-to-weight ratio and is calculated by dividing the ultimate tensile strength of the material by its density. This property is key in weight-critical applications, such as aerospace, where many materials with high strength-to-weight ratios are used (e.g., Ni-based superalloys, Ti-based alloys, Al-based alloys, and composites).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to calculate the strength-to-weight ratio for a given material provided its ultimate tensile strength and density.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePrevious problem: 3 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/8050-stress-strain-properties-3\\\"\u003e\u003cw:r\u003e\u003cw:t\u003equalitative measure of brittleness\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. Next problem: 5 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/8052-stress-strain-properties-5\\\"\u003e\u003cw:r\u003e\u003cw:t\u003estiffness-to-weight ratio\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":8052,"title":"Stress-Strain Properties - 5","description":"Similar to the previous problem, materials may be characterized by their stiffness-to-weight ratio, which is the elastic modulus divided by density. Write a function to calculate this ratio for a material provided its elastic modulus and density.\r\n\r\nPrevious problem: 4 - \u003chttp://www.mathworks.com/matlabcentral/cody/problems/8051-stress-strain-properties-4 strength-to-weight ratio\u003e. Next problem: 6 - \u003chttp://www.mathworks.com/matlabcentral/cody/problems/8053-stress-strain-properties-6 absorbed strain energy\u003e.","description_html":"\u003cp\u003eSimilar to the previous problem, materials may be characterized by their stiffness-to-weight ratio, which is the elastic modulus divided by density. Write a function to calculate this ratio for a material provided its elastic modulus and density.\u003c/p\u003e\u003cp\u003ePrevious problem: 4 - \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/8051-stress-strain-properties-4\"\u003estrength-to-weight ratio\u003c/a\u003e. Next problem: 6 - \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/8053-stress-strain-properties-6\"\u003eabsorbed strain energy\u003c/a\u003e.\u003c/p\u003e","function_template":"function [EtWR] = stress_strain5(E,density)\r\n\r\nEtWR = 1\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile\r\n% in a single direction. Some materials, such as metals are generally\r\n% isotropic, whereas others, like composite are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26;\r\nG = 79.3e9; %Pa\r\nE = 200e9; %Pa\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 0.463; %strain-hardening coefficient\r\nEtWR_corr = 2.548e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 0.974; %strain-hardening coefficient\r\nEtWR_corr = 2.528e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1.845; %strain-hardening coefficient\r\nEtWR_corr = 2.540e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% aluminum alloy (6061-T6)%^\u0026\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 0.325; %strain-hardening coefficient\r\nEtWR_corr = 2.552e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 0.304; %strain-hardening coefficient\r\nEtWR_corr = 1.457e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1.870; %strain-hardening coefficient\r\nEtWR_corr = 2.203e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.1e9; %Pa\r\ndensity = 1.14; %g/cm^3\r\nEtWR_corr = 0.272e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nEtWR_corr = 0.960e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nEtWR_corr = 34.19e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tE = 114e9; %Pa\r\n\t\tdensity = 4.51; %g/cm^3\r\n\t\tEtWR_corr = 2.528e10;\r\n\tcase 2\r\n\t\tE = 68.9e9; %Pa\r\n\t\tdensity = 2.7; %g/cm^3\r\n\t\tEtWR_corr = 2.552e10;\r\n\tcase 3\r\n\t\tE = 200e9; %Pa\r\n\t\tdensity = 7.85; %g/cm^3\r\n\t\tEtWR_corr = 2.548e10;\r\n\tcase 4\r\n\t\tE = 1200e9; %Pa\r\n\t\tdensity = 3.51; %g/cm^3\r\n\t\tEtWR_corr = 34.19e10;\r\nend\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tE = 68.9e9; %Pa\r\n\t\tdensity = 2.7; %g/cm^3\r\n\t\tEtWR_corr = 2.552e10;\r\n\tcase 2\r\n\t\tE = 3.1e9; %Pa\r\n\t\tdensity = 1.14; %g/cm^3\r\n\t\tEtWR_corr = 0.272e10;\r\n\tcase 3\r\n\t\tE = 14.5e9; %Pa\r\n\t\tdensity = 1.51; %g/cm^3\r\n\t\tEtWR_corr = 0.960e10;\r\n\tcase 4\r\n\t\tE = 208e9; %Pa\r\n\t\tdensity = 8.19; %g/cm^3\r\n\t\tEtWR_corr = 2.540e10;\r\nend\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tE = 208e9; %Pa\r\n\t\tdensity = 8.19; %g/cm^3\r\n\t\tEtWR_corr = 2.540e10;\r\n\tcase 2\r\n\t\tE = 463e9; %Pa\r\n\t\tdensity = 21.02; %g/cm^3\r\n\t\tEtWR_corr = 2.203e10;\r\n\tcase 3\r\n\t\tE = 130e9; %Pa\r\n\t\tdensity = 8.92; %g/cm^3\r\n\t\tEtWR_corr = 1.457e10;\r\n\tcase 4\r\n\t\tE = 3.1e9; %Pa\r\n\t\tdensity = 1.14; %g/cm^3\r\n\t\tEtWR_corr = 0.272e10;\r\nend\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":212,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-30T19:40:12.000Z","updated_at":"2026-03-10T20:42:38.000Z","published_at":"2015-03-30T19:40:12.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSimilar to the previous problem, materials may be characterized by their stiffness-to-weight ratio, which is the elastic modulus divided by density. Write a function to calculate this ratio for a material provided its elastic modulus and density.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePrevious problem: 4 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/8051-stress-strain-properties-4\\\"\u003e\u003cw:r\u003e\u003cw:t\u003estrength-to-weight ratio\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. Next problem: 6 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/8053-stress-strain-properties-6\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eabsorbed strain energy\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":8049,"title":"Stress-Strain Properties - 2","description":"The resilience of a material is its ability to resist permanent (or plastic) deformation. The resilience coincides with the elastic region in the figure below and is calculated as the area under the stress-strain curve up to the yield point. Given that the elastic region is presumed to be entirely linear, this area is a triangle. Write a function to calculate the resilience of a material provided its elastic strain and yield stress (yield strength).\r\n\r\n(from quora.com)\r\nPrevious problem: 1 - elastic modulus. Next problem: 3 - qualitative measure of brittleness.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 478px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 332px 239px; transform-origin: 332px 239px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 105px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 52.5px; text-align: left; transform-origin: 309px 52.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe resilience of a material is its ability to resist permanent (or plastic) deformation. The resilience coincides with the elastic region in the figure below and is calculated as the area under the stress-strain curve up to the yield point. Given that the elastic region is presumed to be entirely linear, this area is a triangle. Write a function to calculate the resilience of a material provided its elastic strain and yield stress (yield strength).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 304px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 152px; text-align: center; transform-origin: 309px 152px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline\" src=\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 10.5px; text-align: center; transform-origin: 309px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e(from quora.com)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 10.5px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ePrevious problem: 1 - \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/8048-stress-strain-properties-1\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eelastic modulus\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. Next problem: 3 -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003equalitative measure of brittleness\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [R] = stress_strain2(S_y,e_y)\r\n\r\nR = 1;\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile\r\n% in a single direction. Some materials, such as metals are generally\r\n% isotropic, whereas others, like composite are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26;\r\nG = 79.3e9; %Pa\r\nE = 200e9; %Pa\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 0.463; %strain-hardening coefficient\r\nassert(abs(stress_strain2(S_y,e_y)-1.5625e5)/1.5625e5\u003c5e-2)\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 0.974; %strain-hardening coefficient\r\nassert(abs(stress_strain2(S_y,e_y)-3.0212e6)/3.0212e6\u003c5e-2)\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1.845; %strain-hardening coefficient\r\nassert(abs(stress_strain2(S_y,e_y)-3.29918e6)/3.29918e6\u003c5e-2)\r\n\r\n%% aluminum alloy (6061-T6)%^\u0026\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 0.325; %strain-hardening coefficient\r\nassert(abs(stress_strain2(S_y,e_y)-4.2175e5)/4.2175e5\u003c5e-2)\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 0.304; %strain-hardening coefficient 530MPa\r\nassert(abs(stress_strain2(S_y,e_y)-1.89e4)/1.89e4\u003c5e-2)\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1.870; %strain-hardening coefficient\r\nassert(abs(stress_strain2(S_y,e_y)-1.085725e5)/1.085725e5\u003c5e-2)\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.5e-2; %Pa\r\ndensity = 1.14; %g/cm^3\r\nassert(abs(stress_strain2(S_y,e_y)-1.0865e6)/1.0865e6\u003c5e-2)\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nassert(abs(stress_strain2(S_y,e_y)-1.84e6)/1.84e6\u003c5e-2)\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nassert(abs(stress_strain2(S_y,e_y)-6e5)/6e5\u003c5e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_y = 250e6; %Pa\r\n\t\te_y = 0.00125;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-1.5625e5)/1.5625e5\u003c5e-2)\r\n\tcase 2\r\n\t\tS_y = 82e6; %Pa\r\n\t\te_y = 0.0265;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-1.0865e6)/1.0865e6\u003c5e-2)\r\n\tcase 3\r\n\t\tS_y = 241e6; %Pa\r\n\t\te_y = 0.0035;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-4.2175e5)/4.2175e5\u003c5e-2)\r\n\tcase 4\r\n\t\tS_y = 1172e6; %Pa\r\n\t\te_y = 0.00563;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-3.29918e6)/3.29918e6\u003c5e-2)\r\nend\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_y = 1200e6; %Pa\r\n\t\te_y = 0.001;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-6e5)/6e5\u003c5e-2)\r\n\tcase 2\r\n\t\tS_y = 1172e6; %Pa\r\n\t\te_y = 0.00563;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-3.29918e6)/3.29918e6\u003c5e-2)\r\n\tcase 3\r\n\t\tS_y = 230e6; %Pa\r\n\t\te_y = 0.016;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-1.84e6)/1.84e6\u003c5e-2)\r\n\tcase 4\r\n\t\tS_y = 250e6; %Pa\r\n\t\te_y = 0.00125;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-1.5625e5)/1.5625e5\u003c5e-2)\r\nend\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_y = 830e6; %Pa\r\n\t\te_y = 0.00728;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-3.0212e6)/3.0212e6\u003c5e-2)\r\n\tcase 2\r\n\t\tS_y = 230e6; %Pa\r\n\t\te_y = 0.016;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-1.84e6)/1.84e6\u003c5e-2)\r\n\tcase 3\r\n\t\tS_y = 70e6; %Pa\r\n\t\te_y = 0.00054;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-1.89e4)/1.89e4\u003c5e-2)\r\n\tcase 4\r\n\t\tS_y = 317e6; %Pa\r\n\t\te_y = 0.000685;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-1.085725e5)/1.085725e5\u003c5e-2)\r\nend\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":26769,"edited_by":26769,"edited_at":"2024-03-27T17:41:09.000Z","deleted_by":null,"deleted_at":null,"solvers_count":264,"test_suite_updated_at":"2015-03-30T18:44:33.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-30T18:27:49.000Z","updated_at":"2026-03-31T10:53:49.000Z","published_at":"2015-03-30T18:27:49.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe resilience of a material is its ability to resist permanent (or plastic) deformation. The resilience coincides with the elastic region in the figure below and is calculated as the area under the stress-strain curve up to the yield point. Given that the elastic region is presumed to be entirely linear, this area is a triangle. Write a function to calculate the resilience of a material provided its elastic strain and yield stress (yield strength).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e(from quora.com)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePrevious problem: 1 - \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/8048-stress-strain-properties-1\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eelastic modulus\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. Next problem: 3 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003equalitative measure of brittleness\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"target\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationshipId\":\"rId1\"}]},{\"partUri\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"contentType\":\"image/net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"content\":\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":8050,"title":"Stress-Strain Properties - 3","description":"A brittle material will not exhibit a yield point. In other words, the yield point and failure point coincide. In such cases, the yield strain and failure strain (also known as ultimate strain or elongation) are the same value. On the other hand, ductile materials have a failure strain that is significantly greater than the elastic strain, as shown in the figure below.\r\n\r\n(from quora.com)\r\nWrite a function to determine the qualitative brittleness of the material by calculating the ratio of elastic strain to failure strain. A ratio of one indicates complete brittleness, whereas a ratio close to zero indicates essentially no brittleness.\r\nPrevious problem: 2 - resilience. Next problem: 4 - strength-to-weight ratio.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 529px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 332px 264.5px; transform-origin: 332px 264.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 42px; text-align: left; transform-origin: 309px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eA brittle material will not exhibit a yield point. In other words, the yield point and failure point coincide. In such cases, the yield strain and failure strain (also known as ultimate strain or elongation) are the same value. On the other hand, ductile materials have a failure strain that is significantly greater than the elastic strain, as shown in the figure below.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 304px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 152px; text-align: center; transform-origin: 309px 152px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline\" src=\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 10.5px; text-align: center; transform-origin: 309px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e(from quora.com)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 31.5px; text-align: left; transform-origin: 309px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eWrite a function to determine the qualitative brittleness of the material by calculating the ratio of elastic strain to failure strain. A ratio of one indicates complete brittleness, whereas a ratio close to zero indicates essentially no brittleness.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 10.5px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ePrevious problem: 2 -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/8049-stress-strain-properties-2\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eresilience\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. Next problem: 4 -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003estrength-to-weight ratio\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [BR] = stress_strain3(e_y,e_u)\r\n\r\nBR = 1;\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile\r\n% in a single direction. Some materials, such as metals are generally\r\n% isotropic, whereas others, like composite are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26;\r\nG = 79.3e9; %Pa\r\nE = 200e9; %Pa\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 0.463; %strain-hardening coefficient\r\nBR_corr = 0.003571;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 0.974; %strain-hardening coefficient\r\nBR_corr = 0.052;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1.845; %strain-hardening coefficient\r\nBR_corr = 0.2085;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%% aluminum alloy (6061-T6)%^\u0026\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 0.325; %strain-hardening coefficient\r\nBR_corr = 0.02333;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 0.304; %strain-hardening coefficient\r\nBR_corr = 0.001125;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1.870; %strain-hardening coefficient\r\nBR_corr = 0.002854;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.1e-2; %Pa\r\ndensity = 1.14; %g/cm^3\r\nBR_corr = 0.058889;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nBR_corr = 1.0;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nBR_corr = 1.0;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\te_y = 0.00125;\r\n\t\te_u = 0.35;\r\n\t\tBR_corr = 0.003571;\r\n\tcase 2\r\n\t\te_y = 0.00054;\r\n\t\te_u = 0.48;\r\n\t\tBR_corr = 0.001125;\r\n\tcase 3\r\n\t\te_y = 0.0035;\r\n\t\te_u = 0.15;\r\n\t\tBR_corr = 0.02333;\r\n\tcase 4\r\n\t\te_y = 0.00054;\r\n\t\te_u = 0.48;\r\n\t\tBR_corr = 0.001125;\r\nend\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\te_y = 0.0265;\r\n\t\te_u = 0.45;\r\n\t\tBR_corr = 0.058889;\r\n\tcase 2\r\n\t\te_y = 0.00728;\r\n\t\te_u = 0.14;\r\n\t\tBR_corr = 0.052;\r\n\tcase 3\r\n\t\te_y = 0.00563;\r\n\t\te_u = 0.027;\r\n\t\tBR_corr = 0.2085;\r\n\tcase 4\r\n\t\te_y = 0.016;\r\n\t\te_u = 0.016;\r\n\t\tBR_corr = 1.0;\r\nend\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\te_y = 0.00125;\r\n\t\te_u = 0.35;\r\n\t\tBR_corr = 0.003571;\r\n\tcase 2\r\n\t\te_y = 0.00563;\r\n\t\te_u = 0.027;\r\n\t\tBR_corr = 0.2085;\r\n\tcase 3\r\n\t\te_y = 0.00728;\r\n\t\te_u = 0.14;\r\n\t\tBR_corr = 0.052;\r\n\tcase 4\r\n\t\te_y = 0.00054;\r\n\t\te_u = 0.48;\r\n\t\tBR_corr = 0.001125;\r\nend\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n","published":true,"deleted":false,"likes_count":4,"comments_count":1,"created_by":26769,"edited_by":26769,"edited_at":"2024-03-27T17:42:29.000Z","deleted_by":null,"deleted_at":null,"solvers_count":241,"test_suite_updated_at":"2015-03-30T18:54:51.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-30T18:53:23.000Z","updated_at":"2026-03-31T10:56:32.000Z","published_at":"2015-03-30T18:53:23.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA brittle material will not exhibit a yield point. In other words, the yield point and failure point coincide. In such cases, the yield strain and failure strain (also known as ultimate strain or elongation) are the same value. On the other hand, ductile materials have a failure strain that is significantly greater than the elastic strain, as shown in the figure below.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e(from quora.com)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to determine the qualitative brittleness of the material by calculating the ratio of elastic strain to failure strain. A ratio of one indicates complete brittleness, whereas a ratio close to zero indicates essentially no brittleness.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePrevious problem: 2 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/8049-stress-strain-properties-2\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eresilience\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. Next problem: 4 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003estrength-to-weight ratio\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"target\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationshipId\":\"rId1\"}]},{\"partUri\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"contentType\":\"image/net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"content\":\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60948,"title":"Check p² = 24k + 1, p \u003e 3, the 'golden prime squares' equation","description":"Historical context\r\nIn december 2023, I / Nicolas Douillet was working on prime squares properties and I found* the formula :\r\n\r\n                                                   \r\n\r\nwhich I named the ‘Christmas eve equation'. From it I then wrote :\r\n\r\n\r\n\r\nand I named it the ‘golden prime squares equation’. These two formulas, which are actually the same one, can easily be proven starting from  (note this last formula applies not only to prime numbers). \r\n \r\n* Apparently, I rediscovered it since I found it a posteriori on this excellent French math website :  Site de Gérard Villemin\r\n \r\nProblem statement\r\nYour goal here is to check this formula for the first primes less or equal to a given integer m (input). To do so you will simply compute k in a vector. For the two first cases where the formula doesn’t apply -2 and 3- your function should return 0.\r\n \r\nExamples\r\n\r\nIf m = 20 then k = [0, 0, 1, 2, 5, 7, 12, 15], since 5^2 = 24x1 + 1, 7^2 = 24x2 + 1, 11^2 = 24x5 + 1, etc…\r\n\r\nIf m = 40 then k = [0, 0, 1, 2, 5, 7, 12, 15, 22, 35, 40, 57] ; \r\n \r\nFobidden functions\r\n \r\nregexp\r\nstr2num\r\nassignin\r\n \r\nSee also (my file exchange contributions using this equation) :\r\n\r\nIsprime optimus [FAST]\r\nPrime number spiral (my corresponding contribution on my file exchange)\r\nPrime numbers properties II","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 1019.47px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 509.733px; transform-origin: 408px 509.733px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.725px 8px; transform-origin: 58.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHistorical context\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68.0583px 8px; transform-origin: 68.0583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn december 2023, I / \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/profile/authors/8668631?s_tid=user_nav_comm\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eNicolas Douillet\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 210.05px 8px; transform-origin: 210.05px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e was working on prime squares properties and I found* the formula :\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"251\" height=\"19\" style=\"width: 251px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 99.025px 8px; transform-origin: 99.025px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                                                   \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.7333px 8px; transform-origin: 58.7333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhich I named the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 76.4167px 8px; transform-origin: 76.4167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e‘Christmas eve equation'\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68.0417px 8px; transform-origin: 68.0417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. From it I then wrote :\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"249\" height=\"19\" style=\"width: 249px; height: 19px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 57.95px 8px; transform-origin: 57.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand I named it the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 99.2167px 8px; transform-origin: 99.2167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e‘golden prime squares equation’\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 215.483px 8px; transform-origin: 215.483px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. These two formulas, which are actually the same one, can easily be proven starting from \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"111.5\" height=\"18\" style=\"width: 111.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 182.033px 8px; transform-origin: 182.033px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e (note this last formula applies not only to prime numbers). \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 299.367px 8px; transform-origin: 299.367px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e* Apparently, I rediscovered it since I found it a posteriori on this excellent French math website : \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eSite de Gérard Villemin\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 233.5px 8px; transform-origin: 233.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eYour goal here is to check this formula for the first primes less or equal to a \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 45.4917px 8px; transform-origin: 45.4917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003egiven integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 8.16667px 8px; transform-origin: 8.16667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 97.8417px 8px; transform-origin: 97.8417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(input). To do so you will simply compute \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.5px 8px; transform-origin: 3.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ek\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 211.575px 8px; transform-origin: 211.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in a vector. For the two first cases where the formula doesn’t apply -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e2\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e3\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 94.9083px 8px; transform-origin: 94.9083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e- your function should return 0.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.9833px 8px; transform-origin: 21.9833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 20\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 82.6333px 8px; transform-origin: 82.6333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ek = [0, 0, 1, 2, 5, 7, 12, 15]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.2833px 8px; transform-origin: 18.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e5\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 26.4333px 8px; transform-origin: 26.4333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e^2 = 24x\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e1\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 46.075px 8px; transform-origin: 46.075px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e + 1, 7^2 = 24x\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e2\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.75px 8px; transform-origin: 15.75px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e + 1, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 7.4px 8px; transform-origin: 7.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e11\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 26.4333px 8px; transform-origin: 26.4333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e^2 = 24x\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e5\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 32.0833px 8px; transform-origin: 32.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e + 1, etc…\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.9833px 8px; transform-origin: 21.9833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 40\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 133.575px 8px; transform-origin: 133.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ek = [0, 0, 1, 2, 5, 7, 12, 15, 22, 35, 40, 57] ;\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.9167px 8px; transform-origin: 64.9167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eFobidden functions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 30.65px; transform-origin: 392px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 161.05px 8px; transform-origin: 161.05px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(my file exchange contributions using this equation) \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 30.65px; transform-origin: 392px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/fileexchange/171714-isprime-optimus-fast?s_tid=srchtitle\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-style: italic; \"\u003eIsprime optimus [FAST]\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/fileexchange/171709-prime-numbers-spiral?s_tid=prof_contriblnk\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime number spiral (my corresponding contribution on my file exchange)\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95759\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties II\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function k = golden_prime_squares(m)\r\n\r\n  k = m;\r\n  \r\nend","test_suite":"%%\r\nm = 4;\r\nk_correct = [0, 0];\r\nassert(isequal(golden_prime_squares(m),k_correct))\r\n\r\n%%\r\nm = 20;\r\nk_correct = [0, 0, 1, 2, 5, 7, 12, 15];\r\nassert(isequal(golden_prime_squares(m),k_correct))\r\n\r\n%%\r\nm = 40;\r\nk_correct = [0, 0, 1, 2, 5, 7, 12, 15, 22, 35, 40, 57];\r\nassert(isequal(golden_prime_squares(m),k_correct))\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('golden_prime_squares.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T06:55:57.000Z","deleted_by":null,"deleted_at":null,"solvers_count":22,"test_suite_updated_at":"2025-07-09T05:55:18.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-06-27T16:53:40.000Z","updated_at":"2026-03-11T11:59:56.000Z","published_at":"2025-06-27T17:08:17.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHistorical context\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn december 2023, I / \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/profile/authors/8668631?s_tid=user_nav_comm\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eNicolas Douillet\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e was working on prime squares properties and I found* the formula :\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\forall p \\\\in \\\\mathbb{p}, p \\\\geq 5 \\\\Rightarrow \\\\exists k \\\\in \\\\mathbb{N},  \\\\mathbf{p^2 = 24k + 25}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e                                                   \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhich I named the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e‘Christmas eve equation'\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. From it I then wrote :\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\forall p \\\\in \\\\mathbb{p}, p \\\\geq 5 \\\\Rightarrow \\\\exists k \\\\in \\\\mathbb{N}^*,  \\\\mathbf{p^2 = 24k + 1}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eand I named it the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e‘golden prime squares equation’\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. These two formulas, which are actually the same one, can easily be proven starting from \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep = 6n \\\\pm 1, p \\\\geq 5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e (note this last formula applies not only to prime numbers). \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e* Apparently, I rediscovered it since I found it a posteriori on this excellent French math website : \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eSite de Gérard Villemin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYour goal here is to check this formula for the first primes less or equal to a \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003egiven integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e(input). To do so you will simply compute \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in a vector. For the two first cases where the formula doesn’t apply -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e3\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e- your function should return 0.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 20\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek = [0, 0, 1, 2, 5, 7, 12, 15]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e5\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e^2 = 24x\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e + 1, 7^2 = 24x\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e + 1, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e11\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e^2 = 24x\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e5\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e + 1, etc…\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 40\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek = [0, 0, 1, 2, 5, 7, 12, 15, 22, 35, 40, 57] ;\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eFobidden functions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e(my file exchange contributions using this equation) \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/fileexchange/171714-isprime-optimus-fast?s_tid=srchtitle\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eIsprime optimus [FAST]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/fileexchange/171709-prime-numbers-spiral?s_tid=prof_contriblnk\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime number spiral (my corresponding contribution on my file exchange)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95759\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties II\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60969,"title":"Check p = 4n +/- 1, the generic formula for odd prime numbers","description":"Problem statement\r\n\r\nFor all odd prime number p, there exists a positive integer n, such that p = 4n +/- 1 :\r\n\r\n\r\n\r\nCheck this formula for some given odd primes in a vector by computing n for each p.\r\n\r\nExamples\r\n\r\np = 17 =\u003e n = 4;\r\n\r\np = 19 =\u003e n = 5;\r\n\r\n p = [3, 5, 7, 11, 13, 17, 19] =\u003e n = [1, 1, 2, 3, 3, 4, 5];\r\n\r\np = [3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]\r\n             =\u003e n = [1, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 15, 17, 18, 18, 20, 21, 22, 24];\r\n\r\n\r\nForbidden functions / expressions\r\n\r\nregexp\r\nassignin\r\nstr2num\r\necho\r\n\r\nSee also\r\n\r\nPrime numbers properties I","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 919.467px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 459.733px; transform-origin: 408px 459.733px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 80.1333px 8px; transform-origin: 80.1333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor all odd prime number \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 96.4583px 8px; transform-origin: 96.4583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, there exists a positive integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 8px; transform-origin: 7.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.3333px 8px; transform-origin: 30.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esuch that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 39.675px 8px; transform-origin: 39.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep = 4n +/- 1 :\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"231\" height=\"18\" style=\"width: 231px; height: 18px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 222.1px 8px; transform-origin: 222.1px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCheck this formula for some given odd primes in a vector by computing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.9417px 8px; transform-origin: 15.9417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eeach\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 8px; transform-origin: 7.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e p.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 50.575px 8px; transform-origin: 50.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = 17 =\u0026gt; n = 4;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 50.575px 8px; transform-origin: 50.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = 19 =\u0026gt; n = 5;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 164.367px 8px; transform-origin: 164.367px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [3, 5, 7, 11, 13, 17, 19] =\u0026gt; n = [1, 1, 2, 3, 3, 4, 5];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 280.842px 8px; transform-origin: 280.842px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 287.408px 8px; transform-origin: 287.408px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e             =\u0026gt; n = [1, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 15, 17, 18, 18, 20, 21, 22, 24];\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 114.308px 8px; transform-origin: 114.308px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003eForbidden functions / expressions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties I\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function n = check_odd_primes_gen_formula(p)\r\n  n = p;\r\nend","test_suite":"%%\r\np = 17;\r\nn_correct = 4;\r\nassert(isequal(check_odd_primes_gen_formula(p),n_correct))\r\n\r\n\r\n%%\r\np = 19;\r\nn_correct = 5;\r\nassert(isequal(check_odd_primes_gen_formula(p),n_correct))\r\n\r\n\r\n%%\r\np = [3, 5, 7, 11, 13, 17, 19];\r\nn_correct = [1, 1, 2, 3, 3, 4, 5];\r\nassert(isequal(check_odd_primes_gen_formula(p),n_correct))\r\n\r\n\r\n%%\r\np = [3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97];\r\nn_correct = [1, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 15, 17, 18, 18, 20, 21, 22, 24];\r\nassert(isequal(check_odd_primes_gen_formula(p),n_correct))\r\n\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('check_odd_primes_gen_formula.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-20T05:09:52.000Z","deleted_by":null,"deleted_at":null,"solvers_count":40,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-20T04:24:51.000Z","updated_at":"2026-02-12T07:13:21.000Z","published_at":"2025-07-20T05:09:52.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor all odd prime number \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, there exists a positive integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003esuch that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = 4n +/- 1 :\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\forall p \\\\in \\\\mathbb{P}, p \u0026gt; 2 \\\\Rightarrow \\\\exists n   \\\\in \\\\mathbb{N}, p = 4n \\\\pm 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCheck this formula for some given odd primes in a vector by computing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003eeach\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e p.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = 17 =\u0026gt; n = 4;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = 19 =\u0026gt; n = 5;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [3, 5, 7, 11, 13, 17, 19] =\u0026gt; n = [1, 1, 2, 3, 3, 4, 5];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e             =\u0026gt; n = [1, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 15, 17, 18, 18, 20, 21, 22, 24];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eForbidden functions / expressions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95630\\\"\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003ePrime numbers properties I\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60972,"title":"Check Bertrand-Chebyshev's theorem about prime numbers","description":"Historical context\r\nBertrand's postulate, also later known as Chebyshev's theorem, is a very important theorem in number theory stating that for every integer greater than 1, there always exists at least on prime number between it and its double : \r\n\r\n\r\n\r\nProblem statement\r\nFor a given input n, List the prime number(s) p in the interval ]n; 2n[.\r\n\r\nExamples\r\n\r\nn = 2 =\u003e p = 3;\r\nn = 3 =\u003e p = 5;\r\nn = 7 =\u003e  p = [11, 13];\r\nn = 47 =\u003e p = [53, 59, 61, 67, 71, 73, 79, 83, 89];\r\n\r\nForbidden functions / expressions\r\n\r\nregexp\r\nassignin\r\nstr2num\r\necho\r\n\r\nSee also\r\nPrime numbers properties I","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 697.467px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 348.733px; transform-origin: 408px 348.733px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.725px 8px; transform-origin: 58.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHistorical context\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 385px 8px; transform-origin: 385px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eBertrand's postulate, also later known as Chebyshev's theorem, is a very important theorem in number theory stating that for every integer greater than \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 8px; transform-origin: 7.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e1, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 223.658px 8px; transform-origin: 223.658px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethere always exists at least on prime number between it and its double : \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"223\" height=\"19.5\" style=\"width: 223px; height: 19.5px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 54.0667px 8px; transform-origin: 54.0667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor a given input \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 81.675px 8px; transform-origin: 81.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, List the prime number(s) \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 66.8917px 8px; transform-origin: 66.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in the interval ]n; 2n[.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 46.6833px 8px; transform-origin: 46.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 2 =\u0026gt; p = 3;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 46.6833px 8px; transform-origin: 46.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 3 =\u0026gt; p = 5;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 68.3333px 8px; transform-origin: 68.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 7 =\u0026gt;  p = [11, 13];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 152.467px 8px; transform-origin: 152.467px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 47 =\u0026gt; p = [53, 59, 61, 67, 71, 73, 79, 83, 89];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 114.308px 8px; transform-origin: 114.308px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eForbidden functions / expressions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties I\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function p = check_Bertrand_Chebyshev_theorem(n)\r\n    p = n;\r\nend","test_suite":"%%\r\nn = 2;\r\np_correct = 3;\r\nassert(isequal(check_Bertrand_Chebyshev_theorem(n),p_correct))\r\n\r\n%%\r\nn = 3;\r\np_correct = 5;\r\nassert(isequal(check_Bertrand_Chebyshev_theorem(n),p_correct))\r\n\r\n%%\r\nn = 7;\r\np_correct = [11, 13];\r\nassert(isequal(check_Bertrand_Chebyshev_theorem(n),p_correct))\r\n\r\n%%\r\nn = 11;\r\np_correct = [13, 17, 19];\r\nassert(isequal(check_Bertrand_Chebyshev_theorem(n),p_correct))\r\n\r\n%%\r\nn = 47;\r\np_correct = [53, 59, 61, 67, 71, 73, 79, 83, 89];\r\nassert(isequal(check_Bertrand_Chebyshev_theorem(n),p_correct))\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('check_Bertrand_Chebyshev_theorem.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T07:03:08.000Z","deleted_by":null,"deleted_at":null,"solvers_count":36,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-20T04:28:51.000Z","updated_at":"2026-02-12T07:17:44.000Z","published_at":"2025-07-21T04:30:08.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHistorical context\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBertrand's postulate, also later known as Chebyshev's theorem, is a very important theorem in number theory stating that for every integer greater than \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e1, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethere always exists at least on prime number between it and its double : \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\forall n \\\\in \\\\mathbb{N}^* - \\\\{ 1 \\\\}, \\\\exists p \\\\in \\\\mathbb{P}, \\\\mathbf{n \u0026lt; p \u0026lt; 2n}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor a given input \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, List the prime number(s) \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in the interval ]n; 2n[.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 2 =\u0026gt; p = 3;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 3 =\u0026gt; p = 5;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 7 =\u0026gt;  p = [11, 13];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 47 =\u0026gt; p = [53, 59, 61, 67, 71, 73, 79, 83, 89];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eForbidden functions / expressions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95630\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties I\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60965,"title":"List the near-square primes, p = n² + 1 (Landau's 4th problem)","description":"Historical context\r\n\r\nAt the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as \"unattackable at the present state of mathematics\" and are now known as Landau's problems (source : Wiki)\r\n\r\nProblem statement\r\n\r\nThe 4th Landau's problem aalso known as the 'near square primes' asked whether there are infinitely many primes which are of the form . Your task here is to list such primes below a given input integer m.\r\n\r\nExamples\r\n\r\nm = 20 =\u003e p = [2, 5, 17] ;\r\nm = 200 =\u003e p = [2, 5, 17, 37, 101, 197] ;\r\nm = 2000 =\u003e p = [2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601] ;\r\n\r\nFobidden functions\r\nregexp\r\nstr2num\r\nassignin\r\necho\r\n\r\nSee also\r\n\r\nPrime numbers properties I","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 719.033px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 359.517px; transform-origin: 408px 359.517px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.725px 8px; transform-origin: 58.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHistorical context\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 31.5px; text-align: left; transform-origin: 385px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 375.375px 8px; transform-origin: 375.375px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAt the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as \"unattackable at the present state of mathematics\" and are now known as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 65.825px 8px; transform-origin: 65.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eLandau's problems \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 44.725px 8px; transform-origin: 44.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(source : Wiki)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 376.275px 8px; transform-origin: 376.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe 4th Landau's problem aalso known as the 'near square primes' asked whether there are infinitely many primes which are of the form \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"110.5\" height=\"19\" style=\"width: 110.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 211.775px 8px; transform-origin: 211.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eYour task here is to list such primes below a given input integer\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 30.65px; transform-origin: 392px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 78.575px 8px; transform-origin: 78.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 20 =\u0026gt; p = [2, 5, 17] ;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 125.25px 8px; transform-origin: 125.25px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 200 =\u0026gt; p = [2, 5, 17, 37, 101, 197] ;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 230.275px 8px; transform-origin: 230.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 2000 =\u0026gt; p = [2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601] ;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.9167px 8px; transform-origin: 64.9167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eFobidden functions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e I\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function p = list_the_near_square_primes(m)\r\n  p = m;\r\nend","test_suite":"%%\r\nm = 20;\r\np_correct = [2,5,17];\r\nassert(isequal(list_the_near_square_primes(m),p_correct))\r\n\r\n\r\n%%\r\nm = 200;\r\np_correct = [2,5,17,37,101,197];\r\nassert(isequal(list_the_near_square_primes(m),p_correct))\r\n\r\n\r\n%%\r\nm = 2000;\r\np_correct = [2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601];\r\nassert(isequal(list_the_near_square_primes(m),p_correct))\r\n\r\n\r\n%% Test forbidden functions\r\nfiletext = fileread('list_the_near_square_primes.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);\r\n","published":true,"deleted":false,"likes_count":3,"comments_count":2,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T07:04:20.000Z","deleted_by":null,"deleted_at":null,"solvers_count":55,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-17T04:36:44.000Z","updated_at":"2026-02-13T13:18:04.000Z","published_at":"2025-07-17T05:04:11.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHistorical context\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAt the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as \\\"unattackable at the present state of mathematics\\\" and are now known as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eLandau's problems \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e(source : Wiki)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe 4th Landau's problem aalso known as the 'near square primes' asked whether there are infinitely many primes which are of the form \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep = n^2 + 1, n \\\\in \\\\mathbb{N}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eYour task here is to list such primes below a given input integer\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 20 =\u0026gt; p = [2, 5, 17] ;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 200 =\u0026gt; p = [2, 5, 17, 37, 101, 197] ;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 2000 =\u0026gt; p = [2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601] ;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eFobidden functions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95630\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e I\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":8055,"title":"Stress-Strain Properties - 8","description":"Up to this point, you've calculated some material properties based on tensile stress-strain data. For this problem, you are tasked with writing a function to calculate all of these properties and gather them, along with supplied properties, such as strain values, into an array. You'll be provided a cell array of strings in the function template; you must return an accompanying numerical array that contains all the specified properties. Below is the list of properties for a material, both supplied and calculated, that make up the array (with variable names that have been used):\r\n\r\n* Yield Strength (S_y)\r\n* Yield Strain (e_y)\r\n* Ultimate Strength (S_u)\r\n* Failure Strain (e_u)\r\n* Poisson's Ratio (nu)\r\n* Shear Modulus (G)\r\n* Elastic Modulus (E)\r\n* Density\r\n* Strain-hardening Exponent (sh_exp)\r\n* Strain-hardening Coefficient (sh_coeff)\r\n* Resilience (R)\r\n* Strength-to-weight Ratio (StWR)\r\n* Stiffness-to-weight Ratio (EtWR)\r\n* Absorbed Strain Energy (ASE)\r\n* Toughness (T)\r\n\r\nPrevious problem: 7 - \u003chttp://www.mathworks.com/matlabcentral/cody/problems/8054-stress-strain-properties-7 toughness\u003e.","description_html":"\u003cp\u003eUp to this point, you've calculated some material properties based on tensile stress-strain data. For this problem, you are tasked with writing a function to calculate all of these properties and gather them, along with supplied properties, such as strain values, into an array. You'll be provided a cell array of strings in the function template; you must return an accompanying numerical array that contains all the specified properties. Below is the list of properties for a material, both supplied and calculated, that make up the array (with variable names that have been used):\u003c/p\u003e\u003cul\u003e\u003cli\u003eYield Strength (S_y)\u003c/li\u003e\u003cli\u003eYield Strain (e_y)\u003c/li\u003e\u003cli\u003eUltimate Strength (S_u)\u003c/li\u003e\u003cli\u003eFailure Strain (e_u)\u003c/li\u003e\u003cli\u003ePoisson's Ratio (nu)\u003c/li\u003e\u003cli\u003eShear Modulus (G)\u003c/li\u003e\u003cli\u003eElastic Modulus (E)\u003c/li\u003e\u003cli\u003eDensity\u003c/li\u003e\u003cli\u003eStrain-hardening Exponent (sh_exp)\u003c/li\u003e\u003cli\u003eStrain-hardening Coefficient (sh_coeff)\u003c/li\u003e\u003cli\u003eResilience (R)\u003c/li\u003e\u003cli\u003eStrength-to-weight Ratio (StWR)\u003c/li\u003e\u003cli\u003eStiffness-to-weight Ratio (EtWR)\u003c/li\u003e\u003cli\u003eAbsorbed Strain Energy (ASE)\u003c/li\u003e\u003cli\u003eToughness (T)\u003c/li\u003e\u003c/ul\u003e\u003cp\u003ePrevious problem: 7 - \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/8054-stress-strain-properties-7\"\u003etoughness\u003c/a\u003e.\u003c/p\u003e","function_template":"function [arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff)\r\n\r\narr_descr = {\r\n\t'Yield Strength','Yield Strain','Ultimate Strength','Failure Strain',...\r\n\t'Poisson''s Ratio','Shear Modulus','Elastic Modulus','Density',...\r\n\t'Strain-hardening Exponent','Strain-hardening Coefficient',...\r\n\t'Resilience','Strength-to-weight Ratio','Stiffness-to-weight Ratio',...\r\n\t'Absorbed Strain Energy','Toughness';\r\n};\r\n\r\narr_vals = [\r\n\t1;\r\n];\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile\r\n% in a single direction. Some materials, such as metals are generally\r\n% isotropic, whereas others, like composite are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26; %Poisson's ratio\r\nG = 79.3e9; %Pa (shear modulus)\r\nE = 200e9; %Pa  (elastic modulus)\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 463e6; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t1.5625e5, 5.096e7, 2.548e10, 12.28e7, 12.26e7];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 974e6; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t3.0212e6, 19.96e7, 2.528e10, 12.12e7, 11.82e7];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1845e6; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t3.29918e6, 17.18e7, 2.540e10, 3.535e7, 3.205e7];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n\r\n%% aluminum alloy (6061-T6)%^\u0026\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 325e6; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t4.2175e5, 11.11e7, 2.552e10, 4.321e7, 4.279e7];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 304e6; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t1.89e4, 2.466e7, 1.457e10, 7.342e7, 7.340e7];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1870e6; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t1.085725e5, 5.376e7, 2.203e10, 20.06e7, 20.05e7];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.1e9; %Pa\r\ndensity = 1.14; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t1.0865e6, 7.193e7, 0.272e10, 3.581e7, 3.473e7];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\ndiffs(isnan(diffs)) = 0;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t1.84e6, 15.23e7, 0.960e10, 0.184e7, 0];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\ndiffs(isnan(diffs)) = 0;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t6e5, 34.19e7, 34.19e10, 0.06e7, 0];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\ndiffs(isnan(diffs)) = 0;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":48,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-31T02:36:58.000Z","updated_at":"2026-02-19T09:49:12.000Z","published_at":"2015-03-31T02:36:58.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eUp to this point, you've calculated some material properties based on tensile stress-strain data. For this problem, you are tasked with writing a function to calculate all of these properties and gather them, along with supplied properties, such as strain values, into an array. You'll be provided a cell array of strings in the function template; you must return an accompanying numerical array that contains all the specified properties. Below is the list of properties for a material, both supplied and calculated, that make up the array (with variable names that have been used):\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYield Strength (S_y)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYield Strain (e_y)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eUltimate Strength (S_u)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFailure Strain (e_u)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePoisson's Ratio (nu)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eShear Modulus (G)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eElastic Modulus (E)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDensity\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eStrain-hardening Exponent (sh_exp)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eStrain-hardening Coefficient (sh_coeff)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eResilience (R)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eStrength-to-weight Ratio (StWR)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eStiffness-to-weight Ratio (EtWR)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAbsorbed Strain Energy (ASE)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eToughness (T)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePrevious problem: 7 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/8054-stress-strain-properties-7\\\"\u003e\u003cw:r\u003e\u003cw:t\u003etoughness\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":60949,"title":"Check the integers additive decomposition conjecture","description":"Problem statement\r\n\r\nA conjecture (I rediscovered ?) related to Goldbach's one states that every integer above 2 can be written as the sum of at maximum two prime numbers and the number 1. The goal of this problem is to check this decomposition. Given a positive integer n as an input, your algorithm will return a vector of two primes, [p1, p2], plus potentially the number 1, [1, p1, p2], such that either n = p1 + p2 (case where n is an even number) or n = 1 + p1 + p2 (case where n is an odd number). This p vector will be sorted in ascending order : 1 \u003c p1 \u003c p2 \u003c n. For n = 1 or n = 2 your algorithm should simply return n.\r\n\r\nExamples (check the tests below for more)\r\n\r\nn = 3 =\u003e p = [1, 2] ;\r\nn = 7 =\u003e p = [2, 5] ;\r\nn = 17 =\u003e p = [1, 3, 13] ;\r\nn = 20 =\u003e p = [1; 19] ; % p1 may not be prime in this case\r\nn = 23 =\u003e p = [1, 3, 19] ;\r\nn = 60 =\u003e p = [1, 59] ; % p1 may not be prime in this case\r\nn = 1 =\u003e p = 1 ;\r\nn = 2 =\u003e p = 2;\r\n\r\nTips\r\n\r\nEven if maybe not unique, there is always a solution. If you find a case withouit, at least you will have proven the conjecture to be false ! A simple way to start is to begin with seeking p2, the greater prime before n (even when n is prime itself. Then if the difference between n and this number is a prime number, you just have found p1. Else, add 1 and it should complete the sum.\r\n\r\nForbidden functions\r\n\r\nregexp\r\nstr2num\r\nassignin\r\necho\r\n\r\nSee also\r\n\r\nProblem 60939. Frequencies of prime gaps\r\nPrime numbers properties II","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 956.067px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 478.033px; transform-origin: 408px 478.033px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 105px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 52.5px; text-align: left; transform-origin: 385px 52.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 212.925px 8px; transform-origin: 212.925px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA conjecture (I rediscovered ?) related to Goldbach's one states that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 67.675px 8px; transform-origin: 67.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eevery integer above \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e2 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 96.0417px 8px; transform-origin: 96.0417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003ecan be written as the sum of at maximum two prime numbers and the number \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e1\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e.\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 201.867px 8px; transform-origin: 201.867px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e The goal of this problem is to check this decomposition. Given a positive integer n as an input, your algorithm will return a vector of two primes, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 25.275px 8px; transform-origin: 25.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e[p1, p2],\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 87.525px 8px; transform-origin: 87.525px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e plus potentially the number \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.7083px 8px; transform-origin: 28.7083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e1, [1, p1, p2], \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.3333px 8px; transform-origin: 30.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esuch that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21px 8px; transform-origin: 21px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eeither \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 36.5583px 8px; transform-origin: 36.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = p1 + p2\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.0667px 8px; transform-origin: 40.0667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(case where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18.6667px 8px; transform-origin: 18.6667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is an \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.95px 8px; transform-origin: 15.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eeven\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.95px 8px; transform-origin: 29.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e number) \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7px 8px; transform-origin: 7px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eor\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 50.3667px 8px; transform-origin: 50.3667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 1 + p1 + p2 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.0667px 8px; transform-origin: 40.0667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(case where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 62.2333px 8px; transform-origin: 62.2333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is an odd number). This \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 129.142px 8px; transform-origin: 129.142px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e vector will be sorted in ascending order : \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 52.3083px 8px; transform-origin: 52.3083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e1 \u0026lt; p1 \u0026lt; p2 \u0026lt; n. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.5px 8px; transform-origin: 10.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 43.5667px 8px; transform-origin: 43.5667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e n = 1 or n = 2\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 112.808px 8px; transform-origin: 112.808px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e your algorithm should simply return \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 34.6167px 8px; transform-origin: 34.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 99.5667px 8px; transform-origin: 99.5667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(check the tests below for more)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 163.467px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 81.7333px; transform-origin: 392px 81.7333px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.0833px 8px; transform-origin: 18.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003en = 3 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 8.18333px 8px; transform-origin: 8.18333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e=\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 34.8px 8px; transform-origin: 34.8px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e p = [1, 2] ;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.0833px 8px; transform-origin: 18.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003en = 7 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 8.18333px 8px; transform-origin: 8.18333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e=\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 34.8px 8px; transform-origin: 34.8px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e p = [2, 5] ;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.975px 8px; transform-origin: 21.975px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003en = 17 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 8.18333px 8px; transform-origin: 8.18333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e=\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 46.4667px 8px; transform-origin: 46.4667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e p = [1, 3, 13] ;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 71.1833px 8px; transform-origin: 71.1833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003en = 20 =\u0026gt; p = [1; 19] ; \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 108.908px 8px; transform-origin: 108.908px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e% p1 may not be prime in this case\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.975px 8px; transform-origin: 21.975px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003en = 23 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 8.18333px 8px; transform-origin: 8.18333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e=\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 46.4667px 8px; transform-origin: 46.4667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e p = [1, 3, 19] ;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.975px 8px; transform-origin: 21.975px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003en = 60 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 8.18333px 8px; transform-origin: 8.18333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e=\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 40.6333px 8px; transform-origin: 40.6333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e p = [1, 59] ; \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 108.908px 8px; transform-origin: 108.908px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e% p1 may not be prime in this case\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.0833px 8px; transform-origin: 18.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003en = 1 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 8.18333px 8px; transform-origin: 8.18333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e=\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 22.3583px 8px; transform-origin: 22.3583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e p = 1 ;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.0833px 8px; transform-origin: 18.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003en = 2 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 8.18333px 8px; transform-origin: 8.18333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e=\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 20.4167px 8px; transform-origin: 20.4167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e p = 2;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.2583px 8px; transform-origin: 14.2583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eTips\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 42px; text-align: left; transform-origin: 385px 42px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 383.133px 8px; transform-origin: 383.133px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eEven if maybe not unique, there is always a solution. If you find a case withouit, at least you will have proven the conjecture to be false ! A simple way to start is to begin with seeking \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.78333px 8px; transform-origin: 7.78333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep2\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 80.9px 8px; transform-origin: 80.9px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, the greater prime before \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.0667px 8px; transform-origin: 40.0667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e (even when \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.5583px 8px; transform-origin: 69.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is prime itself. Then if the difference between \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 172.317px 8px; transform-origin: 172.317px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand this number is a prime number, you just have found\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6083px 8px; transform-origin: 13.6083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e p1. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.175px 8px; transform-origin: 29.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eElse, add\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e 1\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 85.575px 8px; transform-origin: 85.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and it should complete the sum.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 67.6417px 8px; transform-origin: 67.6417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eForbidden functions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 40.8667px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 20.4333px; transform-origin: 392px 20.4333px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630/problems/60939\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eProblem 60939. Frequencies of prime gaps\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95759\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties II\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [p] = integer_as_primes_sum(n)\r\n\r\n  p = n;\r\n  \r\nend","test_suite":"%%\r\nn = 1;\r\np_correct = 1;\r\nassert(isequal(integer_as_primes_sum(n),p_correct))\r\n\r\n%%\r\nn = 2;\r\np_correct = 2;\r\nassert(isequal(integer_as_primes_sum(n),p_correct))\r\n\r\n%%\r\nn = 3;\r\np_correct = [1 2];\r\nassert(isequal(integer_as_primes_sum(n),p_correct))\r\n\r\n%%\r\nn = 4;\r\np = integer_as_primes_sum(n);\r\nassert(sum(p) == n \u0026 all(p \u003e 0) \u0026 isequal(floor(p),p) \u0026 numel(find(p == 1)) \u003c 2 \u0026 all(isprime(setdiff(p,1))))\r\n\r\n%%\r\nn = 5;\r\np = integer_as_primes_sum(n);\r\nassert(sum(p) == n \u0026 all(p \u003e 0) \u0026 isequal(floor(p),p) \u0026 numel(find(p == 1)) \u003c 2 \u0026 all(isprime(setdiff(p,1))))\r\n\r\n%%\r\nn = 7;\r\np = integer_as_primes_sum(n);\r\nassert(sum(p) == n \u0026 all(p \u003e 0) \u0026 isequal(floor(p),p) \u0026 numel(find(p == 1)) \u003c 2 \u0026 all(isprime(setdiff(p,1))))\r\n\r\n%%\r\nn = 17;\r\np = integer_as_primes_sum(n);\r\nassert(sum(p) == n \u0026 all(p \u003e 0) \u0026 isequal(floor(p),p) \u0026 numel(find(p == 1)) \u003c 2 \u0026 all(isprime(setdiff(p,1))))\r\n\r\n%%\r\nn = 23;\r\np = integer_as_primes_sum(n);\r\nassert(sum(p) == n \u0026 all(p \u003e 0) \u0026 isequal(floor(p),p) \u0026 numel(find(p == 1)) \u003c 2 \u0026 all(isprime(setdiff(p,1))))\r\n\r\n%%\r\nn = 37;\r\np = integer_as_primes_sum(n);\r\nassert(sum(p) == n \u0026 all(p \u003e 0) \u0026 isequal(floor(p),p) \u0026 numel(find(p == 1)) \u003c 2 \u0026 all(isprime(setdiff(p,1))))\r\n\r\n%%\r\nn = 47;\r\np = integer_as_primes_sum(n);\r\nassert(sum(p) == n \u0026 all(p \u003e 0) \u0026 isequal(floor(p),p) \u0026 numel(find(p == 1)) \u003c 2 \u0026 all(isprime(setdiff(p,1))))\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('integer_as_primes_sum.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":149128,"edited_by":149128,"edited_at":"2025-08-13T04:57:45.000Z","deleted_by":null,"deleted_at":null,"solvers_count":17,"test_suite_updated_at":"2025-08-13T04:57:45.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-06-28T06:09:43.000Z","updated_at":"2026-03-17T10:43:55.000Z","published_at":"2025-06-28T06:44:22.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA conjecture (I rediscovered ?) related to Goldbach's one states that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eevery integer above \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e2 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ecan be written as the sum of at maximum two prime numbers and the number \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e The goal of this problem is to check this decomposition. Given a positive integer n as an input, your algorithm will return a vector of two primes, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[p1, p2],\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e plus potentially the number \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e1, [1, p1, p2], \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esuch that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eeither \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = p1 + p2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e(case where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is an \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eeven\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e number) \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eor\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 1 + p1 + p2 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e(case where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is an odd number). This \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e vector will be sorted in ascending order : \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e1 \u0026lt; p1 \u0026lt; p2 \u0026lt; n. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eFor\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e n = 1 or n = 2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e your algorithm should simply return \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e(check the tests below for more)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 3 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e=\u0026gt;\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e p = [1, 2] ;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 7 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e=\u0026gt;\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e p = [2, 5] ;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 17 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e=\u0026gt;\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e p = [1, 3, 13] ;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 20 =\u0026gt; p = [1; 19] ; \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e% p1 may not be prime in this case\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 23 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e=\u0026gt;\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e p = [1, 3, 19] ;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 60 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e=\u0026gt;\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e p = [1, 59] ; \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e% p1 may not be prime in this case\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 1 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e=\u0026gt;\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e p = 1 ;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 2 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e=\u0026gt;\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e p = 2;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eTips\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEven if maybe not unique, there is always a solution. If you find a case withouit, at least you will have proven the conjecture to be false ! A simple way to start is to begin with seeking \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, the greater prime before \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (even when \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is prime itself. Then if the difference between \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand this number is a prime number, you just have found\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e p1. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eElse, add\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e 1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and it should complete the sum.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eForbidden functions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95630/problems/60939\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 60939. Frequencies of prime gaps\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95759\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties II\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60967,"title":"List primes which are the sum of two consecutive lower primes plus minus one","description":"Problem statement\r\nSome prime numbers can be written as the sum of two consecutive lower primes plus / minus one :\r\n\r\n\r\n\r\nLike this for example, 7 = 3 + 5 - 1, and 11 = 5 + 7 - 1.\r\n\r\nIn a vector, list such prime numbers lower than a given -input- positive integer m.\r\n\r\n\r\nExamples\r\n\r\nm = 50   =\u003e p = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43];\r\nm = 100 =\u003e p = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 59, 61, 67, 79, 83, 89];\r\nm = 200 =\u003e p = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 59, 61, 67, 79, 83, 89, 101, 113, 127, 131, 151, 163, 173, 197, 199];\r\n\r\nFobidden functions\r\n\r\nregexp\r\nstr2num\r\nassignin\r\necho\r\n\r\nSee also\r\n\r\nPrime numbers properties I\r\n","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 813.267px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 406.633px; transform-origin: 408px 406.633px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 306.5px 8px; transform-origin: 306.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSome prime numbers can be written as the sum of two consecutive lower primes plus / minus one :\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 30.8px; border-block-end-color: rgb(60, 60, 60); border-block-start-color: rgb(60, 60, 60); border-bottom-color: rgb(60, 60, 60); border-inline-end-color: rgb(60, 60, 60); border-inline-start-color: rgb(60, 60, 60); border-left-color: rgb(60, 60, 60); border-right-color: rgb(60, 60, 60); border-top-color: rgb(60, 60, 60); caret-color: rgb(60, 60, 60); color: rgb(60, 60, 60); column-rule-color: rgb(60, 60, 60); font-family: Helvetica, Arial, sans-serif; font-size: 20px; font-weight: 700; line-height: 20px; margin-block-end: 5px; margin-block-start: 20px; margin-bottom: 5px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 20px; outline-color: rgb(60, 60, 60); perspective-origin: 385px 15.4px; text-align: left; text-decoration-color: rgb(60, 60, 60); text-emphasis-color: rgb(60, 60, 60); transform-origin: 385px 15.4px; white-space-collapse: preserve; margin-left: 4px; margin-top: 20px; margin-bottom: 5px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-9px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"185.5\" height=\"31\" style=\"width: 185.5px; height: 31px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68.0667px 8px; transform-origin: 68.0667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eLike this for example, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 37.7333px 8px; transform-origin: 37.7333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e7 = 3 + 5 - 1\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 43.05px 8px; transform-origin: 43.05px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e11 = 5 + 7 - 1.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 241.55px 8px; transform-origin: 241.55px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn a vector, list such prime numbers lower than a given -input- positive integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 8px; transform-origin: 7.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 177.225px 8px; transform-origin: 177.225px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 50   =\u0026gt; p = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 258.9px 8px; transform-origin: 258.9px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 100 =\u0026gt; p = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 59, 61, 67, 79, 83, 89];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 40.8667px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 350.95px 8px; transform-origin: 350.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 200 =\u0026gt; p = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 59, 61, 67, 79, 83, 89, 101, 113, 127, 131, 151, 163, 173, 197, 199];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.9167px 8px; transform-origin: 64.9167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eFobidden functions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e I\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function p = prime_as_sum_of_two_consec_primes_pm_1(m)\r\n  p = m;\r\nend","test_suite":"%%\r\nm = 50;\r\np_correct = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43];\r\nassert(isequal(prime_as_sum_of_two_consec_primes_pm_1(m),p_correct))\r\n\r\n\r\n%%\r\nm = 100;\r\np_correct = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 59, 61, 67, 79, 83, 89];\r\nassert(isequal(prime_as_sum_of_two_consec_primes_pm_1(m),p_correct))\r\n\r\n\r\n%%\r\nm = 200;\r\np_correct = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 59, 61, 67, 79, 83, 89, 101, 113, 127, 137, 139, 151, 163, 173, 197, 199];\r\nassert(isequal(prime_as_sum_of_two_consec_primes_pm_1(m),p_correct))\r\n\r\n\r\n%% Test forbidden functions\r\nfiletext = fileread('prime_as_sum_of_two_consec_primes_pm_1.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T07:03:57.000Z","deleted_by":null,"deleted_at":null,"solvers_count":59,"test_suite_updated_at":"2025-07-17T19:19:24.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-17T18:12:11.000Z","updated_at":"2026-03-31T03:46:19.000Z","published_at":"2025-07-17T19:16:00.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSome prime numbers can be written as the sum of two consecutive lower primes plus / minus one :\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"heading\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_m = p_n + p_{n+1} \\\\pm 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eLike this for example, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e7 = 3 + 5 - 1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e11 = 5 + 7 - 1.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn a vector, list such prime numbers lower than a given -input- positive integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 50   =\u0026gt; p = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 100 =\u0026gt; p = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 59, 61, 67, 79, 83, 89];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 200 =\u0026gt; p = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 59, 61, 67, 79, 83, 89, 101, 113, 127, 131, 151, 163, 173, 197, 199];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eFobidden functions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95630\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e I\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":8054,"title":"Stress-Strain Properties - 7","description":"The toughness of a material is technically defined as the plastic strain energy absorbed by the material (the plastic region in the figure below). Practically speaking, it's a measure of how much deformation a material can undergo (or energy it can absorb) before failure.\r\n\r\nWrite a function to calculate the toughness of a material—the absorbed strain energy minus the resilience. This can be accomplished by combining the code written in problem 2 (resilience) and problem 6 (absorbed strain energy). Also, return the fraction of absorbed strain energy that the toughness represents.\r\nPrevious problem: 6 - absorbed strain energy. Next problem: 8 - material properties list.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 499px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 332px 249.5px; transform-origin: 332px 249.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 31.5px; text-align: left; transform-origin: 309px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe toughness of a material is technically defined as the plastic strain energy absorbed by the material (the plastic region in the figure below). Practically speaking, it's a measure of how much deformation a material can undergo (or energy it can absorb) before failure.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 304px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 152px; text-align: center; transform-origin: 309px 152px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline\" src=\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 42px; text-align: left; transform-origin: 309px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eWrite a function to calculate the toughness of a material—the absorbed strain energy minus the resilience. This can be accomplished by combining the code written in\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/8049-stress-strain-properties-2\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eproblem 2\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e (resilience) and\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eproblem 6\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e (absorbed strain energy). Also, return the fraction of absorbed strain energy that the toughness represents.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 10.5px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ePrevious problem: 6 -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eabsorbed strain energy\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. Next problem: 8 -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ematerial properties list\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff)\r\n\r\nT = 1;\r\n\r\nfrac = 0.5;\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile\r\n% in a single direction. Some materials, such as metals, are generally\r\n% isotropic, whereas others, like composites, are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26;\r\nG = 79.3e9; %Pa\r\nE = 200e9; %Pa\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 463e6; %strain-hardening coefficient\r\nT_corr = 12.26e7;\r\nfrac_corr = 0.9987;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\nassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 974e6; %strain-hardening coefficient\r\nT_corr = 11.82e7;\r\nfrac_corr = 0.9751;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\nassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1845e6; %strain-hardening coefficient\r\nT_corr = 3.205e7;\r\nfrac_corr = 0.9067;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\nassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\r\n%% aluminum alloy (6061-T6)\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 325e6; %strain-hardening coefficient\r\nT_corr = 4.279e7;\r\nfrac_corr = 0.9902;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\nassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 304e6; %strain-hardening coefficient\r\nT_corr = 7.340e7;\r\nfrac_corr = 0.9997;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\nassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1870e6; %strain-hardening coefficient\r\nT_corr = 20.05e7;\r\nfrac_corr = 0.9995;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\nassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.1e9; %Pa\r\ndensity = 1.14; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\nT_corr = 3.473e7;\r\nfrac_corr = 0.9697;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\nassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\nT_corr = 0;\r\nfrac_corr = 0;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(isequal(T,T_corr))\r\nassert(isequal(frac,frac_corr))\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\nT_corr = 0;\r\nfrac_corr = 0;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(isequal(T,T_corr))\r\nassert(isequal(frac,frac_corr))\r\n\r\n%%\r\nfor i = 1:30\r\nind = randi(8);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_y = 250e6; %Pa\r\n\t\te_y = 0.00125;\r\n\t\te_u = 0.35;\r\n\t\tsh_exp = 0.14; %strain-hardening exponent\r\n\t\tsh_coeff = 463e6; %strain-hardening coefficient\r\n\t\tT_corr = 12.26e7;\r\n\t\tfrac_corr = 0.9987;\r\n\t\t[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\n\t\tassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\n\t\tassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\tcase 2\r\n\t\tS_y = 830e6; %Pa\r\n\t\te_y = 0.00728;\r\n\t\te_u = 0.14;\r\n\t\tsh_exp = 0.04; %strain-hardening exponent\r\n\t\tsh_coeff = 974e6; %strain-hardening coefficient\r\n\t\tT_corr = 11.82e7;\r\n\t\tfrac_corr = 0.9751;\r\n\t\t[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\n\t\tassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\n\t\tassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\tcase 3\r\n\t\tS_y = 230e6; %Pa\r\n\t\te_y = 0.016;\r\n\t\te_u = 0.016;\r\n\t\tsh_exp = 0; %strain-hardening exponent\r\n\t\tsh_coeff = 0; %strain-hardening coefficient\r\n\t\tT_corr = 0;\r\n\t\tfrac_corr = 0;\r\n\t\t[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\n\t\tassert(isequal(T,T_corr))\r\n\t\tassert(isequal(frac,frac_corr))\r\n\tcase 4\r\n\t\tS_y = 317e6; %Pa\r\n\t\te_y = 0.000685;\r\n\t\te_u = 0.24;\r\n\t\tsh_exp = 0.353; %strain-hardening exponent\r\n\t\tsh_coeff = 1870e6; %strain-hardening coefficient\r\n\t\tT_corr = 20.05e7;\r\n\t\tfrac_corr = 0.9995;\r\n\t\t[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\n\t\tassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\n\t\tassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\tcase 5\r\n\t\tS_y = 70e6; %Pa\r\n\t\te_y = 0.00054;\r\n\t\te_u = 0.48;\r\n\t\tsh_exp = 0.44; %strain-hardening exponent\r\n\t\tsh_coeff = 304e6; %strain-hardening coefficient\r\n\t\tT_corr = 7.340e7;\r\n\t\tfrac_corr = 0.9997;\r\n\tcase 6\r\n\t\tS_y = 1172e6; %Pa\r\n\t\te_y = 0.00563;\r\n\t\te_u = 0.027;\r\n\t\tsh_exp = 0.075; %strain-hardening exponent\r\n\t\tsh_coeff = 1845e6; %strain-hardening coefficient\r\n\t\tT_corr = 3.205e7;\r\n\t\tfrac_corr = 0.9067;\r\n\tcase 7\r\n\t\tS_y = 82e6; %Pa\r\n\t\te_y = 0.0265;\r\n\t\te_u = 0.45;\r\n\t\tsh_exp = 0; %strain-hardening exponent\r\n\t\tsh_coeff = 0; %strain-hardening coefficient\r\n\t\tT_corr = 3.473e7;\r\n\t\tfrac_corr = 0.9697;\r\n\tcase 8\r\n\t\tS_y = 241e6; %Pa\r\n\t\te_y = 0.0035;\r\n\t\te_u = 0.15;\r\n\t\tsh_exp = 0.042; %strain-hardening exponent\r\n\t\tsh_coeff = 325e6; %strain-hardening coefficient\r\n\t\tT_corr = 4.279e7;\r\n\t\tfrac_corr = 0.9902;\r\nend\r\nend % for i = 1:30\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":2,"created_by":26769,"edited_by":26769,"edited_at":"2024-03-27T17:44:32.000Z","deleted_by":null,"deleted_at":null,"solvers_count":88,"test_suite_updated_at":"2021-08-03T17:04:10.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-30T22:03:11.000Z","updated_at":"2026-02-19T09:46:19.000Z","published_at":"2015-03-30T22:03:11.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe toughness of a material is technically defined as the plastic strain energy absorbed by the material (the plastic region in the figure below). Practically speaking, it's a measure of how much deformation a material can undergo (or energy it can absorb) before failure.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"298\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"420\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to calculate the toughness of a material—the absorbed strain energy minus the resilience. This can be accomplished by combining the code written in\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/8049-stress-strain-properties-2\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eproblem 2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e (resilience) and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eproblem 6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e (absorbed strain energy). Also, return the fraction of absorbed strain energy that the toughness represents.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePrevious problem: 6 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eabsorbed strain energy\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. Next problem: 8 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ematerial properties list\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"target\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationshipId\":\"rId1\"}]},{\"partUri\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"contentType\":\"image/net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"content\":\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":8053,"title":"Stress-Strain Properties - 6","description":"The total energy absorbed by a material up to failure in a tensile test is termed the absorbed strain energy. With respect to the figure below, it is the total area of the elastic and plastic regions and can be calculated by integrating the stress-strain curve. As a first approximation, many stress-strain responses can be approximated by:\r\n\r\nwhere K is a strength coefficient, eps_p is the plastic strain, and n is the hardening exponent. Stress as a function of strain can be calculated by creating a strain vector from zero to the ultimate strain and integrating the stress values in that vector.\r\n\r\n(from quora.com)\r\nWrite a function to return the absorbed strain energy for a material provided K and n. If the material does not strain harden, then K and n will be set equal to zero. In these cases, the absorbed strain energy is equal to the resilience (triangular area up to yield point) and any absorbed plastic energy, if applicable, which can be approximated by a rectangle from the yield point to the failure point with those stresses being equal. If the ultimate strain equals the yield strain, that rectangular area is zero.\r\nPrevious problem: 5 - stiffness-to-weight ratio. Next problem: 7 - toughness.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 702px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 332px 351px; transform-origin: 332px 351px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 42px; text-align: left; transform-origin: 309px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe total energy absorbed by a material up to failure in a tensile test is termed the absorbed strain energy. With respect to the figure below, it is the total area of the elastic and plastic regions and can be calculated by integrating the stress-strain curve. As a first approximation, many stress-strain responses can be approximated by:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 29px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 14.5px; text-align: center; transform-origin: 309px 14.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" width=\"73\" height=\"23\" style=\"vertical-align: baseline;width: 73px;height: 23px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 31.5px; text-align: left; transform-origin: 309px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ewhere K is a strength coefficient, eps_p is the plastic strain, and n is the hardening exponent. Stress as a function of strain can be calculated by creating a strain vector from zero to the ultimate strain and integrating the stress values in that vector.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 304px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 152px; text-align: center; transform-origin: 309px 152px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline\" src=\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 10.5px; text-align: center; transform-origin: 309px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e(from quora.com)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 126px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 63px; text-align: left; transform-origin: 309px 63px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eWrite a function to return the absorbed strain energy for a material provided K and n. If the material does not strain harden, then K and n will be set equal to zero. In these cases, the absorbed strain energy is equal to the resilience (triangular area up to yield point) and any absorbed plastic energy, if applicable, which can be approximated by a rectangle from the yield point to the failure point with those stresses being equal. If the ultimate strain equals the yield strain, that rectangular area is zero.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 10.5px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ePrevious problem: 5 -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003estiffness-to-weight ratio\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. Next problem: 7 -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003etoughness\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [ASE] = stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)\r\n\r\nASE = 1;\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile\r\n% in a single direction. Some materials, such as metals are generally\r\n% isotropic, whereas others, like composite are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26;\r\nG = 79.3e9; %Pa\r\nE = 200e9; %Pa\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 463e6; %strain-hardening coefficient\r\nASE_corr = 12.28e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 974e6; %strain-hardening coefficient\r\nASE_corr = 12.12e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1845e6; %strain-hardening coefficient\r\nASE_corr = 3.535e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%% aluminum alloy (6061-T6)%^\u0026\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 325e6; %strain-hardening coefficient\r\nASE_corr = 4.321e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 304e6; %strain-hardening coefficient\r\nASE_corr = 7.342e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1870e6; %strain-hardening coefficient\r\nASE_corr = 20.06e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.1e9; %Pa\r\ndensity = 1.14; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\nASE_corr = 3.581e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\nASE_corr = 0.184e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\nASE_corr = 0.06e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_y = 250e6; %Pa\r\n\t\te_y = 0.00125;\r\n\t\te_u = 0.35;\r\n\t\tsh_exp = 0.14; %strain-hardening exponent\r\n\t\tsh_coeff = 463e6; %strain-hardening coefficient\r\n\t\tASE_corr = 12.28e7;\r\n\tcase 2\r\n\t\tS_y = 82e6; %Pa\r\n\t\te_y = 0.0265;\r\n\t\te_u = 0.45;\r\n\t\tsh_exp = 0; %strain-hardening exponent\r\n\t\tsh_coeff = 0; %strain-hardening coefficient\r\n\t\tASE_corr = 3.581e7;\r\n\tcase 3\r\n\t\tS_y = 241e6; %Pa\r\n\t\te_y = 0.0035;\r\n\t\te_u = 0.15;\r\n\t\tsh_exp = 0.042; %strain-hardening exponent\r\n\t\tsh_coeff = 325e6; %strain-hardening coefficient\r\n\t\tASE_corr = 4.321e7;\r\n\tcase 4\r\n\t\tS_y = 317e6; %Pa\r\n\t\te_y = 0.000685;\r\n\t\te_u = 0.24;\r\n\t\tsh_exp = 0.353; %strain-hardening exponent\r\n\t\tsh_coeff = 1870e6; %strain-hardening coefficient\r\n\t\tASE_corr = 20.06e7;\r\nend\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_y = 830e6; %Pa\r\n\t\te_y = 0.00728;\r\n\t\te_u = 0.14;\r\n\t\tsh_exp = 0.04; %strain-hardening exponent\r\n\t\tsh_coeff = 974e6; %strain-hardening coefficient\r\n\t\tASE_corr = 12.12e7;\r\n\tcase 2\r\n\t\tS_y = 241e6; %Pa\r\n\t\te_y = 0.0035;\r\n\t\te_u = 0.15;\r\n\t\tsh_exp = 0.042; %strain-hardening exponent\r\n\t\tsh_coeff = 325e6; %strain-hardening coefficient\r\n\t\tASE_corr = 4.321e7;\r\n\tcase 3\r\n\t\tS_y = 250e6; %Pa\r\n\t\te_y = 0.00125;\r\n\t\te_u = 0.35;\r\n\t\tsh_exp = 0.14; %strain-hardening exponent\r\n\t\tsh_coeff = 463e6; %strain-hardening coefficient\r\n\t\tASE_corr = 12.28e7;\r\n\tcase 4\r\n\t\tS_y = 70e6; %Pa\r\n\t\te_y = 0.00054;\r\n\t\te_u = 0.48;\r\n\t\tsh_exp = 0.44; %strain-hardening exponent\r\n\t\tsh_coeff = 304e6; %strain-hardening coefficient\r\n\t\tASE_corr = 7.342e7;\r\nend\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_y = 1200e6; %Pa\r\n\t\te_y = 0.001;\r\n\t\te_u = 0.001;\r\n\t\tsh_exp = 0; %strain-hardening exponent\r\n\t\tsh_coeff = 0; %strain-hardening coefficient\r\n\t\tASE_corr = 0.06e7;\r\n\tcase 2\r\n\t\tS_y = 250e6; %Pa\r\n\t\te_y = 0.00125;\r\n\t\te_u = 0.35;\r\n\t\tsh_exp = 0.14; %strain-hardening exponent\r\n\t\tsh_coeff = 463e6; %strain-hardening coefficient\r\n\t\tASE_corr = 12.28e7;\r\n\tcase 3\r\n\t\tS_y = 230e6; %Pa\r\n\t\te_y = 0.016;\r\n\t\te_u = 0.016;\r\n\t\tsh_exp = 0; %strain-hardening exponent\r\n\t\tsh_coeff = 0; %strain-hardening coefficient\r\n\t\tASE_corr = 0.184e7;\r\n\tcase 4\r\n\t\tS_y = 1172e6; %Pa\r\n\t\te_y = 0.00563;\r\n\t\te_u = 0.027;\r\n\t\tsh_exp = 0.075; %strain-hardening exponent\r\n\t\tsh_coeff = 1845e6; %strain-hardening coefficient\r\n\t\tASE_corr = 3.535e7;\r\nend\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":7,"created_by":26769,"edited_by":26769,"edited_at":"2024-03-27T17:39:31.000Z","deleted_by":null,"deleted_at":null,"solvers_count":92,"test_suite_updated_at":"2015-03-30T21:25:46.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-30T20:58:35.000Z","updated_at":"2026-02-19T09:44:40.000Z","published_at":"2015-03-30T21:25:46.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe total energy absorbed by a material up to failure in a tensile test is termed the absorbed strain energy. With respect to the figure below, it is the total area of the elastic and plastic regions and can be calculated by integrating the stress-strain curve. As a first approximation, many stress-strain responses can be approximated by:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"23\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"73\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere K is a strength coefficient, eps_p is the plastic strain, and n is the hardening exponent. Stress as a function of strain can be calculated by creating a strain vector from zero to the ultimate strain and integrating the stress values in that vector.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"298\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"420\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e(from quora.com)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to return the absorbed strain energy for a material provided K and n. If the material does not strain harden, then K and n will be set equal to zero. In these cases, the absorbed strain energy is equal to the resilience (triangular area up to yield point) and any absorbed plastic energy, if applicable, which can be approximated by a rectangle from the yield point to the failure point with those stresses being equal. If the ultimate strain equals the yield strain, that rectangular area is zero.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePrevious problem: 5 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003estiffness-to-weight ratio\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. Next problem: 7 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003etoughness\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"target\":\"/media/image1.png\",\"relationshipId\":\"rId1\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"target\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationshipId\":\"rId2\"}]},{\"partUri\":\"/media/image1.png\",\"contentType\":\"image/png\",\"content\":\"data:image/png;base64,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\",\"relationship\":null},{\"partUri\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"contentType\":\"image/net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"content\":\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60953,"title":"Chek the Delta =  p' - p = 6k gap theorem about arithmetic progressions in the prime number set","description":"Context\r\n \r\nIn the prime numbers set there are some arithmetic progressions (sequences of three or more consecutive prime numbers (p, p’, p’’) equally spaced one to the others by an even number  ).\r\n \r\nOne theorem, which can actually easily be proven from , is that above the sole and unique triplet (3, 5, 7) -with a gap of  then-  all the following progressions are such that \r\n \r\nProblem statement\r\n\r\nFor a given interval [i1, i2], i1 \u003e 7 and i2 \u003e 7 find p the first corresponding arithmetic progression (3 or more consecutive consecutive primes equally spaced) in this interval and check the conjecture equation simply by calculating the integer ratio . \r\n\r\nExamples\r\n                \r\nFor [i1, i2] = [8, 68], p = [47, 53, 59] and k = [1, 1], since this is the first arithmetic progression above 8 and with  here; \r\n \r\nFor [i1, i2] = [180, 228], p = [199, 211, 223], and k = [2, 2], since this is the first arithmetic progression above 180 and with  here;\r\n\r\nFor [i1, i2] = [240, 272], p = [251, 257, 263, 269], and k = [1, 1, 1], since this is the first arithmetic progression above 140 and with  here; \r\n\r\nFor [i1, i2] = [180, 272], p = [199, 211, 223], and k = [2, 2], since this is the first arithmetic progression above 180 and with  here;\r\n\r\nTip\r\n \r\nFirst maybe, train yourself to find the first and last indices of zeros of a first block of consecutive zeros in a vector of integers, eg for u = [1 0 0 0 1 1 0 0 1], j1 = 2 and j2 = 4.\r\n\r\nForbidden functions\r\n \r\n \r\nregexp\r\n \r\nstr2num\r\n \r\nassignin\r\n\r\necho\r\n \r\n \r\nSee also\r\n \r\nProblem 60940. Find the first occurence of a given gap between two consecutive prime numbers\r\nPrime numbers properties I","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 1469.2px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 734.6px; transform-origin: 408px 734.6px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.0583px 8px; transform-origin: 26.0583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eContext\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 248.55px 8px; transform-origin: 248.55px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the prime numbers set there are some arithmetic progressions (sequences of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.45px 8px; transform-origin: 110.45px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003ethree or more consecutive prime numbers\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.825px 8px; transform-origin: 30.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e(p, p’, p’’)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 166.875px 8px; transform-origin: 166.875px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e equally spaced one to the others by an even number \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"134.5\" height=\"18\" style=\"width: 134.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.21667px 8px; transform-origin: 6.21667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e ).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 172.317px 8px; transform-origin: 172.317px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eOne theorem, which can actually easily be proven from \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"236.5\" height=\"19\" style=\"width: 236.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 85.5667px 8px; transform-origin: 85.5667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, is that above the sole and unique triplet \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.1083px 8px; transform-origin: 24.1083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e(3, 5, 7)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 45.8917px 8px; transform-origin: 45.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e -with a gap of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"40\" height=\"18\" style=\"width: 40px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.8917px 8px; transform-origin: 17.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e then-\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 134.208px 8px; transform-origin: 134.208px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eall the following progressions are such that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"97\" height=\"19\" style=\"width: 97px; height: 19px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 31.5px; text-align: left; transform-origin: 385px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 61.4583px 8px; transform-origin: 61.4583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor a given interval \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 75.825px 8px; transform-origin: 75.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e[i1, i2], i1 \u0026gt; 7 and i2 \u0026gt; 7\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.1667px 8px; transform-origin: 15.1667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e find \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4.275px 8px; transform-origin: 4.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 217.05px 8px; transform-origin: 217.05px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e the first corresponding arithmetic progression (3 or more consecutive consecutive primes equally spaced) in this interval and check the conjecture equation simply by calculating the integer ratio \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"57\" height=\"18.5\" style=\"width: 57px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 31.0667px 8px; transform-origin: 31.0667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 40.8667px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 20.4333px; transform-origin: 392px 20.4333px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 12.4417px 8px; transform-origin: 12.4417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 48.4167px 8px; transform-origin: 48.4167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e[i1, i2] = [8, 68]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 48.0333px 8px; transform-origin: 48.0333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [47, 53, 59]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 28.2px 8px; transform-origin: 28.2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ek = [1, 1]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 158.692px 8px; transform-origin: 158.692px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, since this is the first arithmetic progression above \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e8\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 29.95px 8px; transform-origin: 29.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"90\" height=\"18\" style=\"width: 90px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 19.8333px 8px; transform-origin: 19.8333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e here; \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 40.8667px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 20.4333px; transform-origin: 392px 20.4333px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 12.4417px 8px; transform-origin: 12.4417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 60.0917px 8px; transform-origin: 60.0917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e[i1, i2] = [180, 228]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 59.1917px 8px; transform-origin: 59.1917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [199, 211, 223]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 32.0833px 8px; transform-origin: 32.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ek = [2, 2], \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 154.808px 8px; transform-origin: 154.808px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince this is the first arithmetic progression above \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e180\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"97.5\" height=\"18\" style=\"width: 97.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 17.8917px 8px; transform-origin: 17.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e here;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 40.8667px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 20.4333px; transform-origin: 392px 20.4333px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 12.4417px 8px; transform-origin: 12.4417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 60.0917px 8px; transform-origin: 60.0917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e[i1, i2] = [240, 272]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 75.2667px 8px; transform-origin: 75.2667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [251, 257, 263, 269]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 39.8583px 8px; transform-origin: 39.8583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ek = [1, 1, 1], \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 154.808px 8px; transform-origin: 154.808px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince this is the first arithmetic progression above \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e140\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 29.95px 8px; transform-origin: 29.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"90\" height=\"18\" style=\"width: 90px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 19.8333px 8px; transform-origin: 19.8333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e here; \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 40.8667px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 20.4333px; transform-origin: 392px 20.4333px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 12.4417px 8px; transform-origin: 12.4417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 60.0917px 8px; transform-origin: 60.0917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e[i1, i2] = [180, 272]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 59.1917px 8px; transform-origin: 59.1917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [199, 211, 223]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 32.0833px 8px; transform-origin: 32.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ek = [2, 2], \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 154.808px 8px; transform-origin: 154.808px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince this is the first arithmetic progression above \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e180\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMMAAAAkCAYAAAA0Po6QAAAGMUlEQVR4Xu1bTchWRRTOfRHlKhFa1KJA0E0lRDtJECIoSCmJwLAfiDYa1SJCosJw5UIT3UlWO0GCQlokRtqmUrGFLnSRq0SxfT5PzYHj/ebnzNzr+957ORcO78f3zp0558x55vzNu+oef1wDroH/NLDK9eAacA38rwEHg1uCayBowMHgpuAacDC4DbgG7tSAewa3CNeAewa3gYwGHsF374HWg54C/Qn6A3QE9MNENUeZSJsC/5/j87qWpdUz/IVJ9oM+m6hicmw/iC/fB20HrTHIJ+Ofw9jHwvhr+DwFequrcMN8yx7yJhj4AnRvghF+R6BM4eHevAF6FXR/2JMkoFvA8Cwm/R70D+i+KWjEyKMYNQ2YhmCR7wmMOw56KLEG59gAumzkYdnDZG9LfGzGgLF7iK3g8XDYy2/x+WFpH1rA8Asmpevksw30TUlzE/ieinsBdAO0M/BrAcNFjOWJcxR0Mrz3Ij5fDpvAf1nmGYuK6PF5wH0Fugl6GLRFySJ8nsEfG8fCdISPL8M+UvcHQCZPVgsGxlyX1OKMJR8fsVKsrFEuOb0F7CUj/gDvvAN6HvRrZyF6mfMg8Rj0NgetzCxpHMOjd0FPg3QsTe/3YwQQtbazKLEECFyvKqSrFYhegCcFH4kp5+IdZLOsYOApuidj5DQunkp86Kbpfcb80Msxtu4CmzxrWUQGq+3wYDgGehtUEy4ShJ+AGJJZH+r46zCYYVzNu1UdaAr1N+gQiC50d1h07C7TqkgZZwEDPQkT5FyCLfrivNUbU8v0AONpfDEgcGotSy0YmF8yF6nJn7Q3sh4k5PEKSPI9hqtVeY0V3VTAXhDdPRNCPjpcejKjyAH2aaFTWMBgZejfMJAHCKsaU35EFspQEx7TSE+DWGmzAEIDgQctI5E7SqAJJTJs/TR813RAW8EgqPsZi4nrYcj0Ulh8Cief1RCHAoM+TaceSnZzRVZmasrqVkC0AoF7y7BVcjTmCnx0uZsAYeUvybcVDFJ71q6HjJ9VFvYo/q6JCbvGSc8joZfVcGPj+nqpocAg8St7DpZ+hZbFWuIs6anWaFPz6ZyB8qwzntZ6vhIg+gChC1byeA70e2CAPSMBStKrWcFA1DFP6FaOdJm1bygwNzCI52zxCmMDA5NraSj26TGkANEHCLR3rS+GYrF8oRjJWMAgJ1xsU3X2TqZWN5wYpdNt0d8P4RnkpJpD+Kj3eIgScRcQr2ODpTlWkyNou9AHaS7h1qHUikjGAgYaBxtLqX6CXqCvd1i04cfWGwIMnIOPNfkbg9wxHnS/ZAggyBoaEPK/ViDwfQ2GXGioexArxpXAIHlBbgEdT5YaVWPddM1XXzBwYxijxppxU5Bf8yihxZBAkPm7UUWfXM8KhmwfogQGKuMZUClhuoUx0oQbKmlbluH0AQMPho9mAgQ55FJAYJzOgklL0STW1baUXVM2oXOGnP3pcStC2BwYJO61hD4ama3eYeoJ9JBAWHYCLcbK6l7qGgkPjVcawNBNljnHCZC1DxEDhK4m5fI0rdcVdp0DA+MrXjiz3LrsdihbKihTBoMFCNSRpXnEzV4mGGhYv4FSQKAx7wIxH6q9tZyqGpXKrpYoQSpeuYZgthiQAoM02Xh70do51aWrmg6lRdBFjqkNk+Qa92tgMtX+l83uXoJbpFyWtWKJbeq92kpZqXzaFxA6d03lH3LgMnrhjVzTj3uktc3mxVWLFjGGFSepRfOVFu9gXOquDqsBg2zwd+CIPxqJPQ/gnx+Dfqo4WO6qgInJa4DAKWpuhJaAICz1AYSufKWAKpXPaF6R8gy6XNq6MU33Q1oXG+g9fdmLU+b6JqmrzTFWUo2ggdgeZBrt2S0TWptvViAMAQi9VjcnEPmSOXAMDLHruhblxMb0KZe1rtnyHmP0HaDuD1loxAwVeZ9f/4hJ4urUTyO7PIw9bNSX3Kz6szZYxdPW9BG0h6gNx/TPcNdCmAtBIN6g2AdK3mQtlVativFxroGUBmic/F1H7e/BW99r3gkHQ7Pq/MW5acDBMLcddXmaNeBgaFadvzg3DTgY5rajLk+zBhwMzarzF+emAQfD3HbU5WnWgIOhWXX+4tw04GCY2466PM0auA1Dgoc0VXEbjwAAAABJRU5ErkJggg==\" width=\"97.5\" height=\"18\" style=\"width: 97.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 17.8917px 8px; transform-origin: 17.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e here;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.3667px 8px; transform-origin: 10.3667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eTip\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 358.192px 8px; transform-origin: 358.192px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFirst maybe, train yourself to find the first and last indices of zeros of a first block of consecutive zeros in a vector of integers, eg for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 122.325px 8px; transform-origin: 122.325px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eu = [1 0 0 0 1 1 0 0 1], j1 = 2 and j2 = 4.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 67.6417px 8px; transform-origin: 67.6417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eForbidden functions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/problems/60940-find-the-first-occurence-of-a-given-gap-between-two-consecutive-prime-numbers\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eProblem 60940. Find the first occurence of a given gap between two consecutive prime numbers\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties I\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [p, k] = check_the_6k_delta_theorem(i1, i2)\r\n  \r\n    u = i1;\r\n    k = i2;\r\n\r\nend","test_suite":"%%\r\ni1 = 8;\r\ni2 = 68;\r\np_correct = [47, 53, 59];\r\nk_correct = [1, 1];\r\n[p,k] = check_the_6k_delta_theorem(i1,i2);\r\nassert(isequal(p,p_correct) \u0026 isequal(k,k_correct))\r\n\r\n\r\n%%\r\ni1 = 180;\r\ni2 = 228;\r\np_correct = [199, 211, 223];\r\nk_correct = [2, 2];\r\n[p,k] = check_the_6k_delta_theorem(i1,i2);\r\nassert(isequal(p,p_correct) \u0026 isequal(k,k_correct))\r\n\r\n\r\n%%\r\ni1 = 240;\r\ni2 = 272;\r\np_correct = [251, 257, 263, 269];\r\nk_correct = [1, 1, 1];\r\n[p,k] = check_the_6k_delta_theorem(i1,i2);\r\nassert(isequal(p,p_correct) \u0026 isequal(k,k_correct))\r\n\r\n\r\n%%\r\ni1 = 180;\r\ni2 = 272;\r\np_correct = [199, 211, 223];\r\nk_correct = [2, 2];\r\n[p,k] = check_the_6k_delta_theorem(i1,i2);\r\nassert(isequal(p,p_correct) \u0026 isequal(k,k_correct))\r\n\r\n\r\n%% Test forbidden functions\r\nfiletext = fileread('check_the_6k_delta_theorem.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":4,"comments_count":3,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T07:04:42.000Z","deleted_by":null,"deleted_at":null,"solvers_count":34,"test_suite_updated_at":"2025-07-11T05:36:42.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-05T06:40:29.000Z","updated_at":"2026-03-06T14:30:46.000Z","published_at":"2025-07-10T12:03:38.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eContext\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the prime numbers set there are some arithmetic progressions (sequences of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ethree or more consecutive prime numbers\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e(p, p’, p’’)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e equally spaced one to the others by an even number \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mathbf{\\\\Delta = p' - p = p'' - p'}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e ).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOne theorem, which can actually easily be proven from \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\forall p \\\\in \\\\mathbb{P}, p\u0026gt; 3 \\\\Rightarrow \\\\exists n \\\\in \\\\mathbb{N}^*, p = 6n \\\\pm 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, is that above the sole and unique triplet \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e(3, 5, 7)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e -with a gap of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e then-\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eall the following progressions are such that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mathbf{\\\\Delta = 6k}, k \\\\in \\\\mathbb{N}^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor a given interval \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[i1, i2], i1 \u0026gt; 7 and i2 \u0026gt; 7\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e find \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e the first corresponding arithmetic progression (3 or more consecutive consecutive primes equally spaced) in this interval and check the conjecture equation simply by calculating the integer ratio \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mathbf{k = \\\\Delta / 6}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[i1, i2] = [8, 68]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [47, 53, 59]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek = [1, 1]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, since this is the first arithmetic progression above \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e8\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 6 = \\\\mathbf{1} \\\\times 6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e here; \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[i1, i2] = [180, 228]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [199, 211, 223]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek = [2, 2], \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince this is the first arithmetic progression above \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e180\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 12 = \\\\mathbf{2} \\\\times 6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e here;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[i1, i2] = [240, 272]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [251, 257, 263, 269]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek = [1, 1, 1], \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince this is the first arithmetic progression above \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e140\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 6 = \\\\mathbf{1} \\\\times 6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e here; \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[i1, i2] = [180, 272]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [199, 211, 223]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek = [2, 2], \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince this is the first arithmetic progression above \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e180\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 12 = \\\\mathbf{2} \\\\times 6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e here;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eTip\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFirst maybe, train yourself to find the first and last indices of zeros of a first block of consecutive zeros in a vector of integers, eg for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eu = [1 0 0 0 1 1 0 0 1], j1 = 2 and j2 = 4.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eForbidden functions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/problems/60940-find-the-first-occurence-of-a-given-gap-between-two-consecutive-prime-numbers\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 60940. Find the first occurence of a given gap between two consecutive prime numbers\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95630\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties I\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":8048,"title":"Stress-Strain Properties - 1","description":"This is the first in a series of problems regarding mechanics of materials, in particular, material properties drawn from stress-strain responses. A simplified typical stress-strain response is illustrated below (from quora.com):\r\n\r\nThe yield stress is the pressure required to start deformation of the material being tested. The yield point is the point along the response indicated by the yield stress (vertical axis) and the yield strain (horizontal axis). The response of the material up to this point is elastic, meaning that all deformation is reversible. The elastic modulus (E, also known as modulus of elasticity or Young's modulus) is the slope of this line. Write a function to calculate the elastic modulus for a material, provided the elastic strain and yield stress (yield point).\r\nNext problem: 2 - resilience.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 541px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 332px 270.5px; transform-origin: 332px 270.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 31.5px; text-align: left; transform-origin: 309px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThis is the first in a series of problems regarding mechanics of materials, in particular, material properties drawn from stress-strain responses. A simplified typical stress-strain response is illustrated below (from\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e quora.com):\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 304px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 152px; text-align: center; transform-origin: 309px 152px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline\" src=\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 126px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 63px; text-align: left; transform-origin: 309px 63px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe yield stress is the pressure required to start deformation of the material being tested. The yield point is the point along the response indicated by the yield stress (vertical axis) and the yield strain (horizontal axis). The response of the material up to this point is elastic, meaning that all deformation is reversible. The elastic modulus (E, also known as modulus of elasticity or Young's modulus) is the slope of this line. Write a function to calculate the elastic modulus for a material, provided the elastic strain and yield stress (yield point).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 10.5px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eNext problem: 2 -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/8049-stress-strain-properties-2\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eresilience\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [E] = stress_strain1(S_y,e_y)\r\n\r\nE = 1;\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile \r\n% in a single direction. Some materials, such as metals are generally\r\n% isotropic, whereas others, like composite are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26;\r\nG = 79.3e9; %Pa\r\nE = 200e9; %Pa\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 0.463; %strain-hardening coefficient\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 0.974; %strain-hardening coefficient\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1.845; %strain-hardening coefficient\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n\r\n%% aluminum alloy (6061-T6)%^\u0026\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 0.325; %strain-hardening coefficient\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 0.304; %strain-hardening coefficient 530MPa\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1.870; %strain-hardening coefficient\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.1e9; %Pa\r\ndensity = 1.14; %g/cm^3\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":26769,"edited_by":26769,"edited_at":"2024-03-27T17:40:39.000Z","deleted_by":null,"deleted_at":null,"solvers_count":324,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-30T18:09:31.000Z","updated_at":"2026-03-31T10:50:52.000Z","published_at":"2015-03-30T18:09:58.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis is the first in a series of problems regarding mechanics of materials, in particular, material properties drawn from stress-strain responses. A simplified typical stress-strain response is illustrated below (from\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e quora.com):\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe yield stress is the pressure required to start deformation of the material being tested. The yield point is the point along the response indicated by the yield stress (vertical axis) and the yield strain (horizontal axis). The response of the material up to this point is elastic, meaning that all deformation is reversible. The elastic modulus (E, also known as modulus of elasticity or Young's modulus) is the slope of this line. Write a function to calculate the elastic modulus for a material, provided the elastic strain and yield stress (yield point).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNext problem: 2 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/8049-stress-strain-properties-2\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eresilience\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"target\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationshipId\":\"rId1\"}]},{\"partUri\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"contentType\":\"image/net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"content\":\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":8051,"title":"Stress-Strain Properties - 4","description":"A common measure of the ability of a material to carry load per unit mass is termed strength-to-weight ratio and is calculated by dividing the ultimate tensile strength of the material by its density. This property is key in weight-critical applications, such as aerospace, where many materials with high strength-to-weight ratios are used (e.g., Ni-based superalloys, Ti-based alloys, Al-based alloys, and composites).\r\n\r\nWrite a function to calculate the strength-to-weight ratio for a given material provided its ultimate tensile strength and density.\r\n\r\nPrevious problem: 3 - \u003chttp://www.mathworks.com/matlabcentral/cody/problems/8050-stress-strain-properties-3 qualitative measure of brittleness\u003e. Next problem: 5 - \u003chttp://www.mathworks.com/matlabcentral/cody/problems/8052-stress-strain-properties-5 stiffness-to-weight ratio\u003e.","description_html":"\u003cp\u003eA common measure of the ability of a material to carry load per unit mass is termed strength-to-weight ratio and is calculated by dividing the ultimate tensile strength of the material by its density. This property is key in weight-critical applications, such as aerospace, where many materials with high strength-to-weight ratios are used (e.g., Ni-based superalloys, Ti-based alloys, Al-based alloys, and composites).\u003c/p\u003e\u003cp\u003eWrite a function to calculate the strength-to-weight ratio for a given material provided its ultimate tensile strength and density.\u003c/p\u003e\u003cp\u003ePrevious problem: 3 - \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/8050-stress-strain-properties-3\"\u003equalitative measure of brittleness\u003c/a\u003e. Next problem: 5 - \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/8052-stress-strain-properties-5\"\u003estiffness-to-weight ratio\u003c/a\u003e.\u003c/p\u003e","function_template":"function [StWR] = stress_strain4(S_u,density)\r\n\r\nStWR = 1;\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile\r\n% in a single direction. Some materials, such as metals are generally\r\n% isotropic, whereas others, like composite are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26;\r\nG = 79.3e9; %Pa\r\nE = 200e9; %Pa\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 0.463; %strain-hardening coefficient\r\nStWR_corr = 5.096e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 0.974; %strain-hardening coefficient\r\nStWR_corr = 19.96e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1.845; %strain-hardening coefficient\r\nStWR_corr = 17.18e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%% aluminum alloy (6061-T6)%^\u0026\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 0.325; %strain-hardening coefficient\r\nStWR_corr = 11.11e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 0.304; %strain-hardening coefficient\r\nStWR_corr = 2.466e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1.870; %strain-hardening coefficient\r\nStWR_corr = 5.376e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.1e-2; %Pa\r\ndensity = 1.14; %g/cm^3\r\nStWR_corr = 7.193e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nStWR_corr = 15.23e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nStWR_corr = 34.19e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_u = 400e6; %Pa\r\n\t\tdensity = 7.85; %g/cm^3\r\n\t\tStWR_corr = 5.096e7;\r\n\tcase 2\r\n\t\tS_u = 230e6; %Pa\r\n\t\tdensity = 1.51; %g/cm^3\r\n\t\tStWR_corr = 15.23e7;\r\n\tcase 3\r\n\t\tS_u = 1130e6; %Pa\r\n\t\tdensity = 21.02; %g/cm^3\r\n\t\tStWR_corr = 5.376e7;\r\n\tcase 4\r\n\t\tS_u = 1200e6; %Pa\r\n\t\tdensity = 3.51; %g/cm^3\r\n\t\tStWR_corr = 34.19e7;\r\nend\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_u = 300e6; %Pa\r\n\t\tdensity = 2.7; %g/cm^3\r\n\t\tStWR_corr = 11.11e7;\r\n\tcase 2\r\n\t\tS_u = 900e6; %Pa\r\n\t\tdensity = 4.51; %g/cm^3\r\n\t\tStWR_corr = 19.96e7;\r\n\tcase 3\r\n\t\tS_u = 220e6; %Pa\r\n\t\tdensity = 8.92; %g/cm^3\r\n\t\tStWR_corr = 2.466e7;\r\n\tcase 4\r\n\t\tS_u = 230e6; %Pa\r\n\t\tdensity = 1.51; %g/cm^3\r\n\t\tStWR_corr = 15.23e7;\r\nend\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_u = 300e6; %Pa\r\n\t\tdensity = 2.7; %g/cm^3\r\n\t\tStWR_corr = 11.11e7;\r\n\tcase 2\r\n\t\tS_u = 1200e6; %Pa\r\n\t\tdensity = 3.51; %g/cm^3\r\n\t\tStWR_corr = 34.19e7;\r\n\tcase 3\r\n\t\tS_u = 82e6; %Pa\r\n\t\tdensity = 1.14; %g/cm^3\r\n\t\tStWR_corr = 7.193e7;\r\n\tcase 4\r\n\t\tS_u = 900e6; %Pa\r\n\t\tdensity = 4.51; %g/cm^3\r\n\t\tStWR_corr = 19.96e7;\r\nend\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n","published":true,"deleted":false,"likes_count":4,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":222,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-30T19:24:41.000Z","updated_at":"2026-03-10T20:20:32.000Z","published_at":"2015-03-30T19:24:41.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA common measure of the ability of a material to carry load per unit mass is termed strength-to-weight ratio and is calculated by dividing the ultimate tensile strength of the material by its density. This property is key in weight-critical applications, such as aerospace, where many materials with high strength-to-weight ratios are used (e.g., Ni-based superalloys, Ti-based alloys, Al-based alloys, and composites).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to calculate the strength-to-weight ratio for a given material provided its ultimate tensile strength and density.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePrevious problem: 3 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/8050-stress-strain-properties-3\\\"\u003e\u003cw:r\u003e\u003cw:t\u003equalitative measure of brittleness\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. Next problem: 5 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/8052-stress-strain-properties-5\\\"\u003e\u003cw:r\u003e\u003cw:t\u003estiffness-to-weight ratio\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":8052,"title":"Stress-Strain Properties - 5","description":"Similar to the previous problem, materials may be characterized by their stiffness-to-weight ratio, which is the elastic modulus divided by density. Write a function to calculate this ratio for a material provided its elastic modulus and density.\r\n\r\nPrevious problem: 4 - \u003chttp://www.mathworks.com/matlabcentral/cody/problems/8051-stress-strain-properties-4 strength-to-weight ratio\u003e. Next problem: 6 - \u003chttp://www.mathworks.com/matlabcentral/cody/problems/8053-stress-strain-properties-6 absorbed strain energy\u003e.","description_html":"\u003cp\u003eSimilar to the previous problem, materials may be characterized by their stiffness-to-weight ratio, which is the elastic modulus divided by density. Write a function to calculate this ratio for a material provided its elastic modulus and density.\u003c/p\u003e\u003cp\u003ePrevious problem: 4 - \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/8051-stress-strain-properties-4\"\u003estrength-to-weight ratio\u003c/a\u003e. Next problem: 6 - \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/8053-stress-strain-properties-6\"\u003eabsorbed strain energy\u003c/a\u003e.\u003c/p\u003e","function_template":"function [EtWR] = stress_strain5(E,density)\r\n\r\nEtWR = 1\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile\r\n% in a single direction. Some materials, such as metals are generally\r\n% isotropic, whereas others, like composite are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26;\r\nG = 79.3e9; %Pa\r\nE = 200e9; %Pa\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 0.463; %strain-hardening coefficient\r\nEtWR_corr = 2.548e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 0.974; %strain-hardening coefficient\r\nEtWR_corr = 2.528e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1.845; %strain-hardening coefficient\r\nEtWR_corr = 2.540e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% aluminum alloy (6061-T6)%^\u0026\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 0.325; %strain-hardening coefficient\r\nEtWR_corr = 2.552e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 0.304; %strain-hardening coefficient\r\nEtWR_corr = 1.457e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1.870; %strain-hardening coefficient\r\nEtWR_corr = 2.203e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.1e9; %Pa\r\ndensity = 1.14; %g/cm^3\r\nEtWR_corr = 0.272e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nEtWR_corr = 0.960e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nEtWR_corr = 34.19e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tE = 114e9; %Pa\r\n\t\tdensity = 4.51; %g/cm^3\r\n\t\tEtWR_corr = 2.528e10;\r\n\tcase 2\r\n\t\tE = 68.9e9; %Pa\r\n\t\tdensity = 2.7; %g/cm^3\r\n\t\tEtWR_corr = 2.552e10;\r\n\tcase 3\r\n\t\tE = 200e9; %Pa\r\n\t\tdensity = 7.85; %g/cm^3\r\n\t\tEtWR_corr = 2.548e10;\r\n\tcase 4\r\n\t\tE = 1200e9; %Pa\r\n\t\tdensity = 3.51; %g/cm^3\r\n\t\tEtWR_corr = 34.19e10;\r\nend\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tE = 68.9e9; %Pa\r\n\t\tdensity = 2.7; %g/cm^3\r\n\t\tEtWR_corr = 2.552e10;\r\n\tcase 2\r\n\t\tE = 3.1e9; %Pa\r\n\t\tdensity = 1.14; %g/cm^3\r\n\t\tEtWR_corr = 0.272e10;\r\n\tcase 3\r\n\t\tE = 14.5e9; %Pa\r\n\t\tdensity = 1.51; %g/cm^3\r\n\t\tEtWR_corr = 0.960e10;\r\n\tcase 4\r\n\t\tE = 208e9; %Pa\r\n\t\tdensity = 8.19; %g/cm^3\r\n\t\tEtWR_corr = 2.540e10;\r\nend\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tE = 208e9; %Pa\r\n\t\tdensity = 8.19; %g/cm^3\r\n\t\tEtWR_corr = 2.540e10;\r\n\tcase 2\r\n\t\tE = 463e9; %Pa\r\n\t\tdensity = 21.02; %g/cm^3\r\n\t\tEtWR_corr = 2.203e10;\r\n\tcase 3\r\n\t\tE = 130e9; %Pa\r\n\t\tdensity = 8.92; %g/cm^3\r\n\t\tEtWR_corr = 1.457e10;\r\n\tcase 4\r\n\t\tE = 3.1e9; %Pa\r\n\t\tdensity = 1.14; %g/cm^3\r\n\t\tEtWR_corr = 0.272e10;\r\nend\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":212,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-30T19:40:12.000Z","updated_at":"2026-03-10T20:42:38.000Z","published_at":"2015-03-30T19:40:12.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSimilar to the previous problem, materials may be characterized by their stiffness-to-weight ratio, which is the elastic modulus divided by density. Write a function to calculate this ratio for a material provided its elastic modulus and density.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePrevious problem: 4 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/8051-stress-strain-properties-4\\\"\u003e\u003cw:r\u003e\u003cw:t\u003estrength-to-weight ratio\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. Next problem: 6 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/8053-stress-strain-properties-6\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eabsorbed strain energy\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":8049,"title":"Stress-Strain Properties - 2","description":"The resilience of a material is its ability to resist permanent (or plastic) deformation. The resilience coincides with the elastic region in the figure below and is calculated as the area under the stress-strain curve up to the yield point. Given that the elastic region is presumed to be entirely linear, this area is a triangle. Write a function to calculate the resilience of a material provided its elastic strain and yield stress (yield strength).\r\n\r\n(from quora.com)\r\nPrevious problem: 1 - elastic modulus. Next problem: 3 - qualitative measure of brittleness.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 478px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 332px 239px; transform-origin: 332px 239px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 105px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 52.5px; text-align: left; transform-origin: 309px 52.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe resilience of a material is its ability to resist permanent (or plastic) deformation. The resilience coincides with the elastic region in the figure below and is calculated as the area under the stress-strain curve up to the yield point. Given that the elastic region is presumed to be entirely linear, this area is a triangle. Write a function to calculate the resilience of a material provided its elastic strain and yield stress (yield strength).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 304px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 152px; text-align: center; transform-origin: 309px 152px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline\" src=\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 10.5px; text-align: center; transform-origin: 309px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e(from quora.com)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 10.5px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ePrevious problem: 1 - \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/8048-stress-strain-properties-1\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eelastic modulus\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. Next problem: 3 -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003equalitative measure of brittleness\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [R] = stress_strain2(S_y,e_y)\r\n\r\nR = 1;\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile\r\n% in a single direction. Some materials, such as metals are generally\r\n% isotropic, whereas others, like composite are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26;\r\nG = 79.3e9; %Pa\r\nE = 200e9; %Pa\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 0.463; %strain-hardening coefficient\r\nassert(abs(stress_strain2(S_y,e_y)-1.5625e5)/1.5625e5\u003c5e-2)\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 0.974; %strain-hardening coefficient\r\nassert(abs(stress_strain2(S_y,e_y)-3.0212e6)/3.0212e6\u003c5e-2)\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1.845; %strain-hardening coefficient\r\nassert(abs(stress_strain2(S_y,e_y)-3.29918e6)/3.29918e6\u003c5e-2)\r\n\r\n%% aluminum alloy (6061-T6)%^\u0026\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 0.325; %strain-hardening coefficient\r\nassert(abs(stress_strain2(S_y,e_y)-4.2175e5)/4.2175e5\u003c5e-2)\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 0.304; %strain-hardening coefficient 530MPa\r\nassert(abs(stress_strain2(S_y,e_y)-1.89e4)/1.89e4\u003c5e-2)\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1.870; %strain-hardening coefficient\r\nassert(abs(stress_strain2(S_y,e_y)-1.085725e5)/1.085725e5\u003c5e-2)\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.5e-2; %Pa\r\ndensity = 1.14; %g/cm^3\r\nassert(abs(stress_strain2(S_y,e_y)-1.0865e6)/1.0865e6\u003c5e-2)\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nassert(abs(stress_strain2(S_y,e_y)-1.84e6)/1.84e6\u003c5e-2)\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nassert(abs(stress_strain2(S_y,e_y)-6e5)/6e5\u003c5e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_y = 250e6; %Pa\r\n\t\te_y = 0.00125;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-1.5625e5)/1.5625e5\u003c5e-2)\r\n\tcase 2\r\n\t\tS_y = 82e6; %Pa\r\n\t\te_y = 0.0265;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-1.0865e6)/1.0865e6\u003c5e-2)\r\n\tcase 3\r\n\t\tS_y = 241e6; %Pa\r\n\t\te_y = 0.0035;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-4.2175e5)/4.2175e5\u003c5e-2)\r\n\tcase 4\r\n\t\tS_y = 1172e6; %Pa\r\n\t\te_y = 0.00563;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-3.29918e6)/3.29918e6\u003c5e-2)\r\nend\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_y = 1200e6; %Pa\r\n\t\te_y = 0.001;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-6e5)/6e5\u003c5e-2)\r\n\tcase 2\r\n\t\tS_y = 1172e6; %Pa\r\n\t\te_y = 0.00563;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-3.29918e6)/3.29918e6\u003c5e-2)\r\n\tcase 3\r\n\t\tS_y = 230e6; %Pa\r\n\t\te_y = 0.016;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-1.84e6)/1.84e6\u003c5e-2)\r\n\tcase 4\r\n\t\tS_y = 250e6; %Pa\r\n\t\te_y = 0.00125;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-1.5625e5)/1.5625e5\u003c5e-2)\r\nend\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_y = 830e6; %Pa\r\n\t\te_y = 0.00728;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-3.0212e6)/3.0212e6\u003c5e-2)\r\n\tcase 2\r\n\t\tS_y = 230e6; %Pa\r\n\t\te_y = 0.016;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-1.84e6)/1.84e6\u003c5e-2)\r\n\tcase 3\r\n\t\tS_y = 70e6; %Pa\r\n\t\te_y = 0.00054;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-1.89e4)/1.89e4\u003c5e-2)\r\n\tcase 4\r\n\t\tS_y = 317e6; %Pa\r\n\t\te_y = 0.000685;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-1.085725e5)/1.085725e5\u003c5e-2)\r\nend\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":26769,"edited_by":26769,"edited_at":"2024-03-27T17:41:09.000Z","deleted_by":null,"deleted_at":null,"solvers_count":264,"test_suite_updated_at":"2015-03-30T18:44:33.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-30T18:27:49.000Z","updated_at":"2026-03-31T10:53:49.000Z","published_at":"2015-03-30T18:27:49.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe resilience of a material is its ability to resist permanent (or plastic) deformation. The resilience coincides with the elastic region in the figure below and is calculated as the area under the stress-strain curve up to the yield point. Given that the elastic region is presumed to be entirely linear, this area is a triangle. Write a function to calculate the resilience of a material provided its elastic strain and yield stress (yield strength).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e(from quora.com)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePrevious problem: 1 - \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/8048-stress-strain-properties-1\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eelastic modulus\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. Next problem: 3 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003equalitative measure of brittleness\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"target\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationshipId\":\"rId1\"}]},{\"partUri\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"contentType\":\"image/net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"content\":\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":8050,"title":"Stress-Strain Properties - 3","description":"A brittle material will not exhibit a yield point. In other words, the yield point and failure point coincide. In such cases, the yield strain and failure strain (also known as ultimate strain or elongation) are the same value. On the other hand, ductile materials have a failure strain that is significantly greater than the elastic strain, as shown in the figure below.\r\n\r\n(from quora.com)\r\nWrite a function to determine the qualitative brittleness of the material by calculating the ratio of elastic strain to failure strain. A ratio of one indicates complete brittleness, whereas a ratio close to zero indicates essentially no brittleness.\r\nPrevious problem: 2 - resilience. Next problem: 4 - strength-to-weight ratio.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 529px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 332px 264.5px; transform-origin: 332px 264.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 42px; text-align: left; transform-origin: 309px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eA brittle material will not exhibit a yield point. In other words, the yield point and failure point coincide. In such cases, the yield strain and failure strain (also known as ultimate strain or elongation) are the same value. On the other hand, ductile materials have a failure strain that is significantly greater than the elastic strain, as shown in the figure below.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 304px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 152px; text-align: center; transform-origin: 309px 152px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline\" src=\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 10.5px; text-align: center; transform-origin: 309px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e(from quora.com)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 31.5px; text-align: left; transform-origin: 309px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eWrite a function to determine the qualitative brittleness of the material by calculating the ratio of elastic strain to failure strain. A ratio of one indicates complete brittleness, whereas a ratio close to zero indicates essentially no brittleness.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 10.5px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ePrevious problem: 2 -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/8049-stress-strain-properties-2\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eresilience\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. Next problem: 4 -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003estrength-to-weight ratio\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [BR] = stress_strain3(e_y,e_u)\r\n\r\nBR = 1;\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile\r\n% in a single direction. Some materials, such as metals are generally\r\n% isotropic, whereas others, like composite are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26;\r\nG = 79.3e9; %Pa\r\nE = 200e9; %Pa\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 0.463; %strain-hardening coefficient\r\nBR_corr = 0.003571;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 0.974; %strain-hardening coefficient\r\nBR_corr = 0.052;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1.845; %strain-hardening coefficient\r\nBR_corr = 0.2085;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%% aluminum alloy (6061-T6)%^\u0026\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 0.325; %strain-hardening coefficient\r\nBR_corr = 0.02333;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 0.304; %strain-hardening coefficient\r\nBR_corr = 0.001125;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1.870; %strain-hardening coefficient\r\nBR_corr = 0.002854;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.1e-2; %Pa\r\ndensity = 1.14; %g/cm^3\r\nBR_corr = 0.058889;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nBR_corr = 1.0;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nBR_corr = 1.0;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\te_y = 0.00125;\r\n\t\te_u = 0.35;\r\n\t\tBR_corr = 0.003571;\r\n\tcase 2\r\n\t\te_y = 0.00054;\r\n\t\te_u = 0.48;\r\n\t\tBR_corr = 0.001125;\r\n\tcase 3\r\n\t\te_y = 0.0035;\r\n\t\te_u = 0.15;\r\n\t\tBR_corr = 0.02333;\r\n\tcase 4\r\n\t\te_y = 0.00054;\r\n\t\te_u = 0.48;\r\n\t\tBR_corr = 0.001125;\r\nend\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\te_y = 0.0265;\r\n\t\te_u = 0.45;\r\n\t\tBR_corr = 0.058889;\r\n\tcase 2\r\n\t\te_y = 0.00728;\r\n\t\te_u = 0.14;\r\n\t\tBR_corr = 0.052;\r\n\tcase 3\r\n\t\te_y = 0.00563;\r\n\t\te_u = 0.027;\r\n\t\tBR_corr = 0.2085;\r\n\tcase 4\r\n\t\te_y = 0.016;\r\n\t\te_u = 0.016;\r\n\t\tBR_corr = 1.0;\r\nend\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\te_y = 0.00125;\r\n\t\te_u = 0.35;\r\n\t\tBR_corr = 0.003571;\r\n\tcase 2\r\n\t\te_y = 0.00563;\r\n\t\te_u = 0.027;\r\n\t\tBR_corr = 0.2085;\r\n\tcase 3\r\n\t\te_y = 0.00728;\r\n\t\te_u = 0.14;\r\n\t\tBR_corr = 0.052;\r\n\tcase 4\r\n\t\te_y = 0.00054;\r\n\t\te_u = 0.48;\r\n\t\tBR_corr = 0.001125;\r\nend\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n","published":true,"deleted":false,"likes_count":4,"comments_count":1,"created_by":26769,"edited_by":26769,"edited_at":"2024-03-27T17:42:29.000Z","deleted_by":null,"deleted_at":null,"solvers_count":241,"test_suite_updated_at":"2015-03-30T18:54:51.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-30T18:53:23.000Z","updated_at":"2026-03-31T10:56:32.000Z","published_at":"2015-03-30T18:53:23.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA brittle material will not exhibit a yield point. In other words, the yield point and failure point coincide. In such cases, the yield strain and failure strain (also known as ultimate strain or elongation) are the same value. On the other hand, ductile materials have a failure strain that is significantly greater than the elastic strain, as shown in the figure below.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e(from quora.com)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to determine the qualitative brittleness of the material by calculating the ratio of elastic strain to failure strain. A ratio of one indicates complete brittleness, whereas a ratio close to zero indicates essentially no brittleness.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePrevious problem: 2 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/8049-stress-strain-properties-2\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eresilience\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. Next problem: 4 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003estrength-to-weight ratio\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"target\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationshipId\":\"rId1\"}]},{\"partUri\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"contentType\":\"image/net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"content\":\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60948,"title":"Check p² = 24k + 1, p \u003e 3, the 'golden prime squares' equation","description":"Historical context\r\nIn december 2023, I / Nicolas Douillet was working on prime squares properties and I found* the formula :\r\n\r\n                                                   \r\n\r\nwhich I named the ‘Christmas eve equation'. From it I then wrote :\r\n\r\n\r\n\r\nand I named it the ‘golden prime squares equation’. These two formulas, which are actually the same one, can easily be proven starting from  (note this last formula applies not only to prime numbers). \r\n \r\n* Apparently, I rediscovered it since I found it a posteriori on this excellent French math website :  Site de Gérard Villemin\r\n \r\nProblem statement\r\nYour goal here is to check this formula for the first primes less or equal to a given integer m (input). To do so you will simply compute k in a vector. For the two first cases where the formula doesn’t apply -2 and 3- your function should return 0.\r\n \r\nExamples\r\n\r\nIf m = 20 then k = [0, 0, 1, 2, 5, 7, 12, 15], since 5^2 = 24x1 + 1, 7^2 = 24x2 + 1, 11^2 = 24x5 + 1, etc…\r\n\r\nIf m = 40 then k = [0, 0, 1, 2, 5, 7, 12, 15, 22, 35, 40, 57] ; \r\n \r\nFobidden functions\r\n \r\nregexp\r\nstr2num\r\nassignin\r\n \r\nSee also (my file exchange contributions using this equation) :\r\n\r\nIsprime optimus [FAST]\r\nPrime number spiral (my corresponding contribution on my file exchange)\r\nPrime numbers properties II","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 1019.47px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 509.733px; transform-origin: 408px 509.733px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.725px 8px; transform-origin: 58.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHistorical context\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68.0583px 8px; transform-origin: 68.0583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn december 2023, I / \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/profile/authors/8668631?s_tid=user_nav_comm\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eNicolas Douillet\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 210.05px 8px; transform-origin: 210.05px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e was working on prime squares properties and I found* the formula :\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"251\" height=\"19\" style=\"width: 251px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 99.025px 8px; transform-origin: 99.025px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                                                   \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.7333px 8px; transform-origin: 58.7333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhich I named the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 76.4167px 8px; transform-origin: 76.4167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e‘Christmas eve equation'\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68.0417px 8px; transform-origin: 68.0417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. From it I then wrote :\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"249\" height=\"19\" style=\"width: 249px; height: 19px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 57.95px 8px; transform-origin: 57.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand I named it the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 99.2167px 8px; transform-origin: 99.2167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e‘golden prime squares equation’\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 215.483px 8px; transform-origin: 215.483px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. These two formulas, which are actually the same one, can easily be proven starting from \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"111.5\" height=\"18\" style=\"width: 111.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 182.033px 8px; transform-origin: 182.033px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e (note this last formula applies not only to prime numbers). \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 299.367px 8px; transform-origin: 299.367px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e* Apparently, I rediscovered it since I found it a posteriori on this excellent French math website : \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eSite de Gérard Villemin\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 233.5px 8px; transform-origin: 233.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eYour goal here is to check this formula for the first primes less or equal to a \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 45.4917px 8px; transform-origin: 45.4917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003egiven integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 8.16667px 8px; transform-origin: 8.16667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 97.8417px 8px; transform-origin: 97.8417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(input). To do so you will simply compute \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.5px 8px; transform-origin: 3.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ek\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 211.575px 8px; transform-origin: 211.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in a vector. For the two first cases where the formula doesn’t apply -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e2\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e3\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 94.9083px 8px; transform-origin: 94.9083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e- your function should return 0.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.9833px 8px; transform-origin: 21.9833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 20\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 82.6333px 8px; transform-origin: 82.6333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ek = [0, 0, 1, 2, 5, 7, 12, 15]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.2833px 8px; transform-origin: 18.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e5\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 26.4333px 8px; transform-origin: 26.4333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e^2 = 24x\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e1\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 46.075px 8px; transform-origin: 46.075px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e + 1, 7^2 = 24x\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e2\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.75px 8px; transform-origin: 15.75px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e + 1, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 7.4px 8px; transform-origin: 7.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e11\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 26.4333px 8px; transform-origin: 26.4333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e^2 = 24x\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e5\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 32.0833px 8px; transform-origin: 32.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e + 1, etc…\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.9833px 8px; transform-origin: 21.9833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 40\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.6167px 8px; transform-origin: 13.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethen\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 133.575px 8px; transform-origin: 133.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ek = [0, 0, 1, 2, 5, 7, 12, 15, 22, 35, 40, 57] ;\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.9167px 8px; transform-origin: 64.9167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eFobidden functions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 30.65px; transform-origin: 392px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 161.05px 8px; transform-origin: 161.05px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(my file exchange contributions using this equation) \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 30.65px; transform-origin: 392px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/fileexchange/171714-isprime-optimus-fast?s_tid=srchtitle\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-style: italic; \"\u003eIsprime optimus [FAST]\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/fileexchange/171709-prime-numbers-spiral?s_tid=prof_contriblnk\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime number spiral (my corresponding contribution on my file exchange)\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95759\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties II\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function k = golden_prime_squares(m)\r\n\r\n  k = m;\r\n  \r\nend","test_suite":"%%\r\nm = 4;\r\nk_correct = [0, 0];\r\nassert(isequal(golden_prime_squares(m),k_correct))\r\n\r\n%%\r\nm = 20;\r\nk_correct = [0, 0, 1, 2, 5, 7, 12, 15];\r\nassert(isequal(golden_prime_squares(m),k_correct))\r\n\r\n%%\r\nm = 40;\r\nk_correct = [0, 0, 1, 2, 5, 7, 12, 15, 22, 35, 40, 57];\r\nassert(isequal(golden_prime_squares(m),k_correct))\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('golden_prime_squares.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T06:55:57.000Z","deleted_by":null,"deleted_at":null,"solvers_count":22,"test_suite_updated_at":"2025-07-09T05:55:18.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-06-27T16:53:40.000Z","updated_at":"2026-03-11T11:59:56.000Z","published_at":"2025-06-27T17:08:17.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHistorical context\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn december 2023, I / \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/profile/authors/8668631?s_tid=user_nav_comm\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eNicolas Douillet\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e was working on prime squares properties and I found* the formula :\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\forall p \\\\in \\\\mathbb{p}, p \\\\geq 5 \\\\Rightarrow \\\\exists k \\\\in \\\\mathbb{N},  \\\\mathbf{p^2 = 24k + 25}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e                                                   \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhich I named the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e‘Christmas eve equation'\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. From it I then wrote :\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\forall p \\\\in \\\\mathbb{p}, p \\\\geq 5 \\\\Rightarrow \\\\exists k \\\\in \\\\mathbb{N}^*,  \\\\mathbf{p^2 = 24k + 1}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eand I named it the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e‘golden prime squares equation’\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. These two formulas, which are actually the same one, can easily be proven starting from \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep = 6n \\\\pm 1, p \\\\geq 5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e (note this last formula applies not only to prime numbers). \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e* Apparently, I rediscovered it since I found it a posteriori on this excellent French math website : \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eSite de Gérard Villemin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYour goal here is to check this formula for the first primes less or equal to a \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003egiven integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e(input). To do so you will simply compute \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in a vector. For the two first cases where the formula doesn’t apply -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e3\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e- your function should return 0.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 20\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek = [0, 0, 1, 2, 5, 7, 12, 15]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e5\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e^2 = 24x\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e + 1, 7^2 = 24x\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e + 1, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e11\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e^2 = 24x\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e5\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e + 1, etc…\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 40\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethen\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek = [0, 0, 1, 2, 5, 7, 12, 15, 22, 35, 40, 57] ;\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eFobidden functions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e(my file exchange contributions using this equation) \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/fileexchange/171714-isprime-optimus-fast?s_tid=srchtitle\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eIsprime optimus [FAST]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/fileexchange/171709-prime-numbers-spiral?s_tid=prof_contriblnk\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime number spiral (my corresponding contribution on my file exchange)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95759\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties II\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60969,"title":"Check p = 4n +/- 1, the generic formula for odd prime numbers","description":"Problem statement\r\n\r\nFor all odd prime number p, there exists a positive integer n, such that p = 4n +/- 1 :\r\n\r\n\r\n\r\nCheck this formula for some given odd primes in a vector by computing n for each p.\r\n\r\nExamples\r\n\r\np = 17 =\u003e n = 4;\r\n\r\np = 19 =\u003e n = 5;\r\n\r\n p = [3, 5, 7, 11, 13, 17, 19] =\u003e n = [1, 1, 2, 3, 3, 4, 5];\r\n\r\np = [3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]\r\n             =\u003e n = [1, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 15, 17, 18, 18, 20, 21, 22, 24];\r\n\r\n\r\nForbidden functions / expressions\r\n\r\nregexp\r\nassignin\r\nstr2num\r\necho\r\n\r\nSee also\r\n\r\nPrime numbers properties I","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 919.467px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 459.733px; transform-origin: 408px 459.733px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 80.1333px 8px; transform-origin: 80.1333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor all odd prime number \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 96.4583px 8px; transform-origin: 96.4583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, there exists a positive integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 8px; transform-origin: 7.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.3333px 8px; transform-origin: 30.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esuch that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 39.675px 8px; transform-origin: 39.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep = 4n +/- 1 :\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"231\" height=\"18\" style=\"width: 231px; height: 18px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 222.1px 8px; transform-origin: 222.1px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCheck this formula for some given odd primes in a vector by computing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.9417px 8px; transform-origin: 15.9417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eeach\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 8px; transform-origin: 7.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e p.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 50.575px 8px; transform-origin: 50.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = 17 =\u0026gt; n = 4;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 50.575px 8px; transform-origin: 50.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = 19 =\u0026gt; n = 5;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 164.367px 8px; transform-origin: 164.367px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [3, 5, 7, 11, 13, 17, 19] =\u0026gt; n = [1, 1, 2, 3, 3, 4, 5];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 280.842px 8px; transform-origin: 280.842px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 287.408px 8px; transform-origin: 287.408px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e             =\u0026gt; n = [1, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 15, 17, 18, 18, 20, 21, 22, 24];\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 114.308px 8px; transform-origin: 114.308px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003eForbidden functions / expressions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties I\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function n = check_odd_primes_gen_formula(p)\r\n  n = p;\r\nend","test_suite":"%%\r\np = 17;\r\nn_correct = 4;\r\nassert(isequal(check_odd_primes_gen_formula(p),n_correct))\r\n\r\n\r\n%%\r\np = 19;\r\nn_correct = 5;\r\nassert(isequal(check_odd_primes_gen_formula(p),n_correct))\r\n\r\n\r\n%%\r\np = [3, 5, 7, 11, 13, 17, 19];\r\nn_correct = [1, 1, 2, 3, 3, 4, 5];\r\nassert(isequal(check_odd_primes_gen_formula(p),n_correct))\r\n\r\n\r\n%%\r\np = [3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97];\r\nn_correct = [1, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 15, 17, 18, 18, 20, 21, 22, 24];\r\nassert(isequal(check_odd_primes_gen_formula(p),n_correct))\r\n\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('check_odd_primes_gen_formula.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-20T05:09:52.000Z","deleted_by":null,"deleted_at":null,"solvers_count":40,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-20T04:24:51.000Z","updated_at":"2026-02-12T07:13:21.000Z","published_at":"2025-07-20T05:09:52.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor all odd prime number \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, there exists a positive integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003esuch that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = 4n +/- 1 :\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\forall p \\\\in \\\\mathbb{P}, p \u0026gt; 2 \\\\Rightarrow \\\\exists n   \\\\in \\\\mathbb{N}, p = 4n \\\\pm 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCheck this formula for some given odd primes in a vector by computing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003eeach\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e p.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = 17 =\u0026gt; n = 4;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = 19 =\u0026gt; n = 5;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [3, 5, 7, 11, 13, 17, 19] =\u0026gt; n = [1, 1, 2, 3, 3, 4, 5];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e             =\u0026gt; n = [1, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 15, 17, 18, 18, 20, 21, 22, 24];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eForbidden functions / expressions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95630\\\"\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003ePrime numbers properties I\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60972,"title":"Check Bertrand-Chebyshev's theorem about prime numbers","description":"Historical context\r\nBertrand's postulate, also later known as Chebyshev's theorem, is a very important theorem in number theory stating that for every integer greater than 1, there always exists at least on prime number between it and its double : \r\n\r\n\r\n\r\nProblem statement\r\nFor a given input n, List the prime number(s) p in the interval ]n; 2n[.\r\n\r\nExamples\r\n\r\nn = 2 =\u003e p = 3;\r\nn = 3 =\u003e p = 5;\r\nn = 7 =\u003e  p = [11, 13];\r\nn = 47 =\u003e p = [53, 59, 61, 67, 71, 73, 79, 83, 89];\r\n\r\nForbidden functions / expressions\r\n\r\nregexp\r\nassignin\r\nstr2num\r\necho\r\n\r\nSee also\r\nPrime numbers properties I","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 697.467px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 348.733px; transform-origin: 408px 348.733px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.725px 8px; transform-origin: 58.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHistorical context\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 385px 8px; transform-origin: 385px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eBertrand's postulate, also later known as Chebyshev's theorem, is a very important theorem in number theory stating that for every integer greater than \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 8px; transform-origin: 7.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e1, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 223.658px 8px; transform-origin: 223.658px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethere always exists at least on prime number between it and its double : \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"223\" height=\"19.5\" style=\"width: 223px; height: 19.5px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 54.0667px 8px; transform-origin: 54.0667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor a given input \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 81.675px 8px; transform-origin: 81.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, List the prime number(s) \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 66.8917px 8px; transform-origin: 66.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in the interval ]n; 2n[.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 46.6833px 8px; transform-origin: 46.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 2 =\u0026gt; p = 3;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 46.6833px 8px; transform-origin: 46.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 3 =\u0026gt; p = 5;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 68.3333px 8px; transform-origin: 68.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 7 =\u0026gt;  p = [11, 13];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 152.467px 8px; transform-origin: 152.467px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 47 =\u0026gt; p = [53, 59, 61, 67, 71, 73, 79, 83, 89];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 114.308px 8px; transform-origin: 114.308px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eForbidden functions / expressions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties I\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function p = check_Bertrand_Chebyshev_theorem(n)\r\n    p = n;\r\nend","test_suite":"%%\r\nn = 2;\r\np_correct = 3;\r\nassert(isequal(check_Bertrand_Chebyshev_theorem(n),p_correct))\r\n\r\n%%\r\nn = 3;\r\np_correct = 5;\r\nassert(isequal(check_Bertrand_Chebyshev_theorem(n),p_correct))\r\n\r\n%%\r\nn = 7;\r\np_correct = [11, 13];\r\nassert(isequal(check_Bertrand_Chebyshev_theorem(n),p_correct))\r\n\r\n%%\r\nn = 11;\r\np_correct = [13, 17, 19];\r\nassert(isequal(check_Bertrand_Chebyshev_theorem(n),p_correct))\r\n\r\n%%\r\nn = 47;\r\np_correct = [53, 59, 61, 67, 71, 73, 79, 83, 89];\r\nassert(isequal(check_Bertrand_Chebyshev_theorem(n),p_correct))\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('check_Bertrand_Chebyshev_theorem.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T07:03:08.000Z","deleted_by":null,"deleted_at":null,"solvers_count":36,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-20T04:28:51.000Z","updated_at":"2026-02-12T07:17:44.000Z","published_at":"2025-07-21T04:30:08.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHistorical context\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBertrand's postulate, also later known as Chebyshev's theorem, is a very important theorem in number theory stating that for every integer greater than \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e1, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethere always exists at least on prime number between it and its double : \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\forall n \\\\in \\\\mathbb{N}^* - \\\\{ 1 \\\\}, \\\\exists p \\\\in \\\\mathbb{P}, \\\\mathbf{n \u0026lt; p \u0026lt; 2n}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor a given input \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, List the prime number(s) \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in the interval ]n; 2n[.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 2 =\u0026gt; p = 3;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 3 =\u0026gt; p = 5;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 7 =\u0026gt;  p = [11, 13];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 47 =\u0026gt; p = [53, 59, 61, 67, 71, 73, 79, 83, 89];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eForbidden functions / expressions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95630\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties I\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60965,"title":"List the near-square primes, p = n² + 1 (Landau's 4th problem)","description":"Historical context\r\n\r\nAt the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as \"unattackable at the present state of mathematics\" and are now known as Landau's problems (source : Wiki)\r\n\r\nProblem statement\r\n\r\nThe 4th Landau's problem aalso known as the 'near square primes' asked whether there are infinitely many primes which are of the form . Your task here is to list such primes below a given input integer m.\r\n\r\nExamples\r\n\r\nm = 20 =\u003e p = [2, 5, 17] ;\r\nm = 200 =\u003e p = [2, 5, 17, 37, 101, 197] ;\r\nm = 2000 =\u003e p = [2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601] ;\r\n\r\nFobidden functions\r\nregexp\r\nstr2num\r\nassignin\r\necho\r\n\r\nSee also\r\n\r\nPrime numbers properties I","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 719.033px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 359.517px; transform-origin: 408px 359.517px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.725px 8px; transform-origin: 58.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHistorical context\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 31.5px; text-align: left; transform-origin: 385px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 375.375px 8px; transform-origin: 375.375px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAt the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as \"unattackable at the present state of mathematics\" and are now known as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 65.825px 8px; transform-origin: 65.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eLandau's problems \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 44.725px 8px; transform-origin: 44.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(source : Wiki)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 376.275px 8px; transform-origin: 376.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe 4th Landau's problem aalso known as the 'near square primes' asked whether there are infinitely many primes which are of the form \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"110.5\" height=\"19\" style=\"width: 110.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 211.775px 8px; transform-origin: 211.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eYour task here is to list such primes below a given input integer\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 30.65px; transform-origin: 392px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 78.575px 8px; transform-origin: 78.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 20 =\u0026gt; p = [2, 5, 17] ;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 125.25px 8px; transform-origin: 125.25px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 200 =\u0026gt; p = [2, 5, 17, 37, 101, 197] ;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 230.275px 8px; transform-origin: 230.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 2000 =\u0026gt; p = [2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601] ;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.9167px 8px; transform-origin: 64.9167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eFobidden functions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e I\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function p = list_the_near_square_primes(m)\r\n  p = m;\r\nend","test_suite":"%%\r\nm = 20;\r\np_correct = [2,5,17];\r\nassert(isequal(list_the_near_square_primes(m),p_correct))\r\n\r\n\r\n%%\r\nm = 200;\r\np_correct = [2,5,17,37,101,197];\r\nassert(isequal(list_the_near_square_primes(m),p_correct))\r\n\r\n\r\n%%\r\nm = 2000;\r\np_correct = [2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601];\r\nassert(isequal(list_the_near_square_primes(m),p_correct))\r\n\r\n\r\n%% Test forbidden functions\r\nfiletext = fileread('list_the_near_square_primes.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);\r\n","published":true,"deleted":false,"likes_count":3,"comments_count":2,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T07:04:20.000Z","deleted_by":null,"deleted_at":null,"solvers_count":55,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-17T04:36:44.000Z","updated_at":"2026-02-13T13:18:04.000Z","published_at":"2025-07-17T05:04:11.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHistorical context\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAt the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as \\\"unattackable at the present state of mathematics\\\" and are now known as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eLandau's problems \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e(source : Wiki)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe 4th Landau's problem aalso known as the 'near square primes' asked whether there are infinitely many primes which are of the form \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep = n^2 + 1, n \\\\in \\\\mathbb{N}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eYour task here is to list such primes below a given input integer\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 20 =\u0026gt; p = [2, 5, 17] ;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 200 =\u0026gt; p = [2, 5, 17, 37, 101, 197] ;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 2000 =\u0026gt; p = [2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601] ;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eFobidden functions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95630\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e I\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":8055,"title":"Stress-Strain Properties - 8","description":"Up to this point, you've calculated some material properties based on tensile stress-strain data. For this problem, you are tasked with writing a function to calculate all of these properties and gather them, along with supplied properties, such as strain values, into an array. You'll be provided a cell array of strings in the function template; you must return an accompanying numerical array that contains all the specified properties. Below is the list of properties for a material, both supplied and calculated, that make up the array (with variable names that have been used):\r\n\r\n* Yield Strength (S_y)\r\n* Yield Strain (e_y)\r\n* Ultimate Strength (S_u)\r\n* Failure Strain (e_u)\r\n* Poisson's Ratio (nu)\r\n* Shear Modulus (G)\r\n* Elastic Modulus (E)\r\n* Density\r\n* Strain-hardening Exponent (sh_exp)\r\n* Strain-hardening Coefficient (sh_coeff)\r\n* Resilience (R)\r\n* Strength-to-weight Ratio (StWR)\r\n* Stiffness-to-weight Ratio (EtWR)\r\n* Absorbed Strain Energy (ASE)\r\n* Toughness (T)\r\n\r\nPrevious problem: 7 - \u003chttp://www.mathworks.com/matlabcentral/cody/problems/8054-stress-strain-properties-7 toughness\u003e.","description_html":"\u003cp\u003eUp to this point, you've calculated some material properties based on tensile stress-strain data. For this problem, you are tasked with writing a function to calculate all of these properties and gather them, along with supplied properties, such as strain values, into an array. You'll be provided a cell array of strings in the function template; you must return an accompanying numerical array that contains all the specified properties. Below is the list of properties for a material, both supplied and calculated, that make up the array (with variable names that have been used):\u003c/p\u003e\u003cul\u003e\u003cli\u003eYield Strength (S_y)\u003c/li\u003e\u003cli\u003eYield Strain (e_y)\u003c/li\u003e\u003cli\u003eUltimate Strength (S_u)\u003c/li\u003e\u003cli\u003eFailure Strain (e_u)\u003c/li\u003e\u003cli\u003ePoisson's Ratio (nu)\u003c/li\u003e\u003cli\u003eShear Modulus (G)\u003c/li\u003e\u003cli\u003eElastic Modulus (E)\u003c/li\u003e\u003cli\u003eDensity\u003c/li\u003e\u003cli\u003eStrain-hardening Exponent (sh_exp)\u003c/li\u003e\u003cli\u003eStrain-hardening Coefficient (sh_coeff)\u003c/li\u003e\u003cli\u003eResilience (R)\u003c/li\u003e\u003cli\u003eStrength-to-weight Ratio (StWR)\u003c/li\u003e\u003cli\u003eStiffness-to-weight Ratio (EtWR)\u003c/li\u003e\u003cli\u003eAbsorbed Strain Energy (ASE)\u003c/li\u003e\u003cli\u003eToughness (T)\u003c/li\u003e\u003c/ul\u003e\u003cp\u003ePrevious problem: 7 - \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/8054-stress-strain-properties-7\"\u003etoughness\u003c/a\u003e.\u003c/p\u003e","function_template":"function [arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff)\r\n\r\narr_descr = {\r\n\t'Yield Strength','Yield Strain','Ultimate Strength','Failure Strain',...\r\n\t'Poisson''s Ratio','Shear Modulus','Elastic Modulus','Density',...\r\n\t'Strain-hardening Exponent','Strain-hardening Coefficient',...\r\n\t'Resilience','Strength-to-weight Ratio','Stiffness-to-weight Ratio',...\r\n\t'Absorbed Strain Energy','Toughness';\r\n};\r\n\r\narr_vals = [\r\n\t1;\r\n];\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile\r\n% in a single direction. Some materials, such as metals are generally\r\n% isotropic, whereas others, like composite are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26; %Poisson's ratio\r\nG = 79.3e9; %Pa (shear modulus)\r\nE = 200e9; %Pa  (elastic modulus)\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 463e6; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t1.5625e5, 5.096e7, 2.548e10, 12.28e7, 12.26e7];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 974e6; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t3.0212e6, 19.96e7, 2.528e10, 12.12e7, 11.82e7];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1845e6; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t3.29918e6, 17.18e7, 2.540e10, 3.535e7, 3.205e7];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n\r\n%% aluminum alloy (6061-T6)%^\u0026\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 325e6; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t4.2175e5, 11.11e7, 2.552e10, 4.321e7, 4.279e7];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 304e6; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t1.89e4, 2.466e7, 1.457e10, 7.342e7, 7.340e7];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1870e6; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t1.085725e5, 5.376e7, 2.203e10, 20.06e7, 20.05e7];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.1e9; %Pa\r\ndensity = 1.14; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t1.0865e6, 7.193e7, 0.272e10, 3.581e7, 3.473e7];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\ndiffs(isnan(diffs)) = 0;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t1.84e6, 15.23e7, 0.960e10, 0.184e7, 0];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\ndiffs(isnan(diffs)) = 0;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t6e5, 34.19e7, 34.19e10, 0.06e7, 0];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\ndiffs(isnan(diffs)) = 0;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":48,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-31T02:36:58.000Z","updated_at":"2026-02-19T09:49:12.000Z","published_at":"2015-03-31T02:36:58.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eUp to this point, you've calculated some material properties based on tensile stress-strain data. For this problem, you are tasked with writing a function to calculate all of these properties and gather them, along with supplied properties, such as strain values, into an array. You'll be provided a cell array of strings in the function template; you must return an accompanying numerical array that contains all the specified properties. Below is the list of properties for a material, both supplied and calculated, that make up the array (with variable names that have been used):\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYield Strength (S_y)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYield Strain (e_y)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eUltimate Strength (S_u)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFailure Strain (e_u)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePoisson's Ratio (nu)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eShear Modulus (G)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eElastic Modulus (E)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDensity\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eStrain-hardening Exponent (sh_exp)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eStrain-hardening Coefficient (sh_coeff)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eResilience (R)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eStrength-to-weight Ratio (StWR)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eStiffness-to-weight Ratio (EtWR)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAbsorbed Strain Energy (ASE)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eToughness (T)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePrevious problem: 7 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/8054-stress-strain-properties-7\\\"\u003e\u003cw:r\u003e\u003cw:t\u003etoughness\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":60949,"title":"Check the integers additive decomposition conjecture","description":"Problem statement\r\n\r\nA conjecture (I rediscovered ?) related to Goldbach's one states that every integer above 2 can be written as the sum of at maximum two prime numbers and the number 1. The goal of this problem is to check this decomposition. Given a positive integer n as an input, your algorithm will return a vector of two primes, [p1, p2], plus potentially the number 1, [1, p1, p2], such that either n = p1 + p2 (case where n is an even number) or n = 1 + p1 + p2 (case where n is an odd number). This p vector will be sorted in ascending order : 1 \u003c p1 \u003c p2 \u003c n. For n = 1 or n = 2 your algorithm should simply return n.\r\n\r\nExamples (check the tests below for more)\r\n\r\nn = 3 =\u003e p = [1, 2] ;\r\nn = 7 =\u003e p = [2, 5] ;\r\nn = 17 =\u003e p = [1, 3, 13] ;\r\nn = 20 =\u003e p = [1; 19] ; % p1 may not be prime in this case\r\nn = 23 =\u003e p = [1, 3, 19] ;\r\nn = 60 =\u003e p = [1, 59] ; % p1 may not be prime in this case\r\nn = 1 =\u003e p = 1 ;\r\nn = 2 =\u003e p = 2;\r\n\r\nTips\r\n\r\nEven if maybe not unique, there is always a solution. If you find a case withouit, at least you will have proven the conjecture to be false ! A simple way to start is to begin with seeking p2, the greater prime before n (even when n is prime itself. Then if the difference between n and this number is a prime number, you just have found p1. Else, add 1 and it should complete the sum.\r\n\r\nForbidden functions\r\n\r\nregexp\r\nstr2num\r\nassignin\r\necho\r\n\r\nSee also\r\n\r\nProblem 60939. Frequencies of prime gaps\r\nPrime numbers properties II","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 956.067px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 478.033px; transform-origin: 408px 478.033px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 105px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 52.5px; text-align: left; transform-origin: 385px 52.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 212.925px 8px; transform-origin: 212.925px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA conjecture (I rediscovered ?) related to Goldbach's one states that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 67.675px 8px; transform-origin: 67.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eevery integer above \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e2 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 96.0417px 8px; transform-origin: 96.0417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003ecan be written as the sum of at maximum two prime numbers and the number \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e1\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e.\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 201.867px 8px; transform-origin: 201.867px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e The goal of this problem is to check this decomposition. Given a positive integer n as an input, your algorithm will return a vector of two primes, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 25.275px 8px; transform-origin: 25.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e[p1, p2],\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 87.525px 8px; transform-origin: 87.525px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e plus potentially the number \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.7083px 8px; transform-origin: 28.7083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e1, [1, p1, p2], \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.3333px 8px; transform-origin: 30.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esuch that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21px 8px; transform-origin: 21px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eeither \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 36.5583px 8px; transform-origin: 36.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = p1 + p2\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.0667px 8px; transform-origin: 40.0667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(case where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18.6667px 8px; transform-origin: 18.6667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is an \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.95px 8px; transform-origin: 15.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eeven\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.95px 8px; transform-origin: 29.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e number) \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7px 8px; transform-origin: 7px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eor\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 50.3667px 8px; transform-origin: 50.3667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003en = 1 + p1 + p2 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.0667px 8px; transform-origin: 40.0667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(case where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 62.2333px 8px; transform-origin: 62.2333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is an odd number). This \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 129.142px 8px; transform-origin: 129.142px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e vector will be sorted in ascending order : \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 52.3083px 8px; transform-origin: 52.3083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e1 \u0026lt; p1 \u0026lt; p2 \u0026lt; n. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.5px 8px; transform-origin: 10.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 43.5667px 8px; transform-origin: 43.5667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e n = 1 or n = 2\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 112.808px 8px; transform-origin: 112.808px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e your algorithm should simply return \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 34.6167px 8px; transform-origin: 34.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 99.5667px 8px; transform-origin: 99.5667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e(check the tests below for more)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 163.467px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 81.7333px; transform-origin: 392px 81.7333px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.0833px 8px; transform-origin: 18.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003en = 3 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 8.18333px 8px; transform-origin: 8.18333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e=\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 34.8px 8px; transform-origin: 34.8px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e p = [1, 2] ;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.0833px 8px; transform-origin: 18.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003en = 7 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 8.18333px 8px; transform-origin: 8.18333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e=\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 34.8px 8px; transform-origin: 34.8px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e p = [2, 5] ;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.975px 8px; transform-origin: 21.975px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003en = 17 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 8.18333px 8px; transform-origin: 8.18333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e=\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 46.4667px 8px; transform-origin: 46.4667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e p = [1, 3, 13] ;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 71.1833px 8px; transform-origin: 71.1833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003en = 20 =\u0026gt; p = [1; 19] ; \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 108.908px 8px; transform-origin: 108.908px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e% p1 may not be prime in this case\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.975px 8px; transform-origin: 21.975px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003en = 23 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 8.18333px 8px; transform-origin: 8.18333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e=\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 46.4667px 8px; transform-origin: 46.4667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e p = [1, 3, 19] ;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.975px 8px; transform-origin: 21.975px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003en = 60 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 8.18333px 8px; transform-origin: 8.18333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e=\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 40.6333px 8px; transform-origin: 40.6333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e p = [1, 59] ; \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 108.908px 8px; transform-origin: 108.908px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e% p1 may not be prime in this case\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.0833px 8px; transform-origin: 18.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003en = 1 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 8.18333px 8px; transform-origin: 8.18333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e=\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 22.3583px 8px; transform-origin: 22.3583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e p = 1 ;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 18.0833px 8px; transform-origin: 18.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003en = 2 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 8.18333px 8px; transform-origin: 8.18333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e=\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 20.4167px 8px; transform-origin: 20.4167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e p = 2;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.2583px 8px; transform-origin: 14.2583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eTips\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 42px; text-align: left; transform-origin: 385px 42px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 383.133px 8px; transform-origin: 383.133px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eEven if maybe not unique, there is always a solution. If you find a case withouit, at least you will have proven the conjecture to be false ! A simple way to start is to begin with seeking \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.78333px 8px; transform-origin: 7.78333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep2\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 80.9px 8px; transform-origin: 80.9px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, the greater prime before \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.0667px 8px; transform-origin: 40.0667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e (even when \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.5583px 8px; transform-origin: 69.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is prime itself. Then if the difference between \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 172.317px 8px; transform-origin: 172.317px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eand this number is a prime number, you just have found\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6083px 8px; transform-origin: 13.6083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e p1. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.175px 8px; transform-origin: 29.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eElse, add\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e 1\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 85.575px 8px; transform-origin: 85.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and it should complete the sum.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 67.6417px 8px; transform-origin: 67.6417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eForbidden functions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 40.8667px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 20.4333px; transform-origin: 392px 20.4333px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630/problems/60939\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eProblem 60939. Frequencies of prime gaps\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95759\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties II\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [p] = integer_as_primes_sum(n)\r\n\r\n  p = n;\r\n  \r\nend","test_suite":"%%\r\nn = 1;\r\np_correct = 1;\r\nassert(isequal(integer_as_primes_sum(n),p_correct))\r\n\r\n%%\r\nn = 2;\r\np_correct = 2;\r\nassert(isequal(integer_as_primes_sum(n),p_correct))\r\n\r\n%%\r\nn = 3;\r\np_correct = [1 2];\r\nassert(isequal(integer_as_primes_sum(n),p_correct))\r\n\r\n%%\r\nn = 4;\r\np = integer_as_primes_sum(n);\r\nassert(sum(p) == n \u0026 all(p \u003e 0) \u0026 isequal(floor(p),p) \u0026 numel(find(p == 1)) \u003c 2 \u0026 all(isprime(setdiff(p,1))))\r\n\r\n%%\r\nn = 5;\r\np = integer_as_primes_sum(n);\r\nassert(sum(p) == n \u0026 all(p \u003e 0) \u0026 isequal(floor(p),p) \u0026 numel(find(p == 1)) \u003c 2 \u0026 all(isprime(setdiff(p,1))))\r\n\r\n%%\r\nn = 7;\r\np = integer_as_primes_sum(n);\r\nassert(sum(p) == n \u0026 all(p \u003e 0) \u0026 isequal(floor(p),p) \u0026 numel(find(p == 1)) \u003c 2 \u0026 all(isprime(setdiff(p,1))))\r\n\r\n%%\r\nn = 17;\r\np = integer_as_primes_sum(n);\r\nassert(sum(p) == n \u0026 all(p \u003e 0) \u0026 isequal(floor(p),p) \u0026 numel(find(p == 1)) \u003c 2 \u0026 all(isprime(setdiff(p,1))))\r\n\r\n%%\r\nn = 23;\r\np = integer_as_primes_sum(n);\r\nassert(sum(p) == n \u0026 all(p \u003e 0) \u0026 isequal(floor(p),p) \u0026 numel(find(p == 1)) \u003c 2 \u0026 all(isprime(setdiff(p,1))))\r\n\r\n%%\r\nn = 37;\r\np = integer_as_primes_sum(n);\r\nassert(sum(p) == n \u0026 all(p \u003e 0) \u0026 isequal(floor(p),p) \u0026 numel(find(p == 1)) \u003c 2 \u0026 all(isprime(setdiff(p,1))))\r\n\r\n%%\r\nn = 47;\r\np = integer_as_primes_sum(n);\r\nassert(sum(p) == n \u0026 all(p \u003e 0) \u0026 isequal(floor(p),p) \u0026 numel(find(p == 1)) \u003c 2 \u0026 all(isprime(setdiff(p,1))))\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('integer_as_primes_sum.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":149128,"edited_by":149128,"edited_at":"2025-08-13T04:57:45.000Z","deleted_by":null,"deleted_at":null,"solvers_count":17,"test_suite_updated_at":"2025-08-13T04:57:45.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-06-28T06:09:43.000Z","updated_at":"2026-03-17T10:43:55.000Z","published_at":"2025-06-28T06:44:22.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA conjecture (I rediscovered ?) related to Goldbach's one states that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eevery integer above \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e2 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ecan be written as the sum of at maximum two prime numbers and the number \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e The goal of this problem is to check this decomposition. Given a positive integer n as an input, your algorithm will return a vector of two primes, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[p1, p2],\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e plus potentially the number \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e1, [1, p1, p2], \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esuch that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eeither \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = p1 + p2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e(case where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is an \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eeven\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e number) \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eor\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 1 + p1 + p2 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e(case where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is an odd number). This \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e vector will be sorted in ascending order : \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e1 \u0026lt; p1 \u0026lt; p2 \u0026lt; n. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eFor\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e n = 1 or n = 2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e your algorithm should simply return \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e(check the tests below for more)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 3 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e=\u0026gt;\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e p = [1, 2] ;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 7 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e=\u0026gt;\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e p = [2, 5] ;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 17 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e=\u0026gt;\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e p = [1, 3, 13] ;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 20 =\u0026gt; p = [1; 19] ; \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e% p1 may not be prime in this case\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 23 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e=\u0026gt;\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e p = [1, 3, 19] ;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 60 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e=\u0026gt;\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e p = [1, 59] ; \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e% p1 may not be prime in this case\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 1 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e=\u0026gt;\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e p = 1 ;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 2 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e=\u0026gt;\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e p = 2;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eTips\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEven if maybe not unique, there is always a solution. If you find a case withouit, at least you will have proven the conjecture to be false ! A simple way to start is to begin with seeking \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, the greater prime before \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (even when \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is prime itself. Then if the difference between \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eand this number is a prime number, you just have found\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e p1. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eElse, add\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e 1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and it should complete the sum.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eForbidden functions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95630/problems/60939\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 60939. Frequencies of prime gaps\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95759\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties II\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60967,"title":"List primes which are the sum of two consecutive lower primes plus minus one","description":"Problem statement\r\nSome prime numbers can be written as the sum of two consecutive lower primes plus / minus one :\r\n\r\n\r\n\r\nLike this for example, 7 = 3 + 5 - 1, and 11 = 5 + 7 - 1.\r\n\r\nIn a vector, list such prime numbers lower than a given -input- positive integer m.\r\n\r\n\r\nExamples\r\n\r\nm = 50   =\u003e p = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43];\r\nm = 100 =\u003e p = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 59, 61, 67, 79, 83, 89];\r\nm = 200 =\u003e p = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 59, 61, 67, 79, 83, 89, 101, 113, 127, 131, 151, 163, 173, 197, 199];\r\n\r\nFobidden functions\r\n\r\nregexp\r\nstr2num\r\nassignin\r\necho\r\n\r\nSee also\r\n\r\nPrime numbers properties I\r\n","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 813.267px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 406.633px; transform-origin: 408px 406.633px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 306.5px 8px; transform-origin: 306.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSome prime numbers can be written as the sum of two consecutive lower primes plus / minus one :\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 30.8px; border-block-end-color: rgb(60, 60, 60); border-block-start-color: rgb(60, 60, 60); border-bottom-color: rgb(60, 60, 60); border-inline-end-color: rgb(60, 60, 60); border-inline-start-color: rgb(60, 60, 60); border-left-color: rgb(60, 60, 60); border-right-color: rgb(60, 60, 60); border-top-color: rgb(60, 60, 60); caret-color: rgb(60, 60, 60); color: rgb(60, 60, 60); column-rule-color: rgb(60, 60, 60); font-family: Helvetica, Arial, sans-serif; font-size: 20px; font-weight: 700; line-height: 20px; margin-block-end: 5px; margin-block-start: 20px; margin-bottom: 5px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 20px; outline-color: rgb(60, 60, 60); perspective-origin: 385px 15.4px; text-align: left; text-decoration-color: rgb(60, 60, 60); text-emphasis-color: rgb(60, 60, 60); transform-origin: 385px 15.4px; white-space-collapse: preserve; margin-left: 4px; margin-top: 20px; margin-bottom: 5px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-9px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"185.5\" height=\"31\" style=\"width: 185.5px; height: 31px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68.0667px 8px; transform-origin: 68.0667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eLike this for example, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 37.7333px 8px; transform-origin: 37.7333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e7 = 3 + 5 - 1\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 43.05px 8px; transform-origin: 43.05px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e11 = 5 + 7 - 1.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 241.55px 8px; transform-origin: 241.55px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn a vector, list such prime numbers lower than a given -input- positive integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 8px; transform-origin: 7.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 177.225px 8px; transform-origin: 177.225px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 50   =\u0026gt; p = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 258.9px 8px; transform-origin: 258.9px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 100 =\u0026gt; p = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 59, 61, 67, 79, 83, 89];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 40.8667px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 350.95px 8px; transform-origin: 350.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003em = 200 =\u0026gt; p = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 59, 61, 67, 79, 83, 89, 101, 113, 127, 131, 151, 163, 173, 197, 199];\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.9167px 8px; transform-origin: 64.9167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eFobidden functions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e I\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function p = prime_as_sum_of_two_consec_primes_pm_1(m)\r\n  p = m;\r\nend","test_suite":"%%\r\nm = 50;\r\np_correct = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43];\r\nassert(isequal(prime_as_sum_of_two_consec_primes_pm_1(m),p_correct))\r\n\r\n\r\n%%\r\nm = 100;\r\np_correct = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 59, 61, 67, 79, 83, 89];\r\nassert(isequal(prime_as_sum_of_two_consec_primes_pm_1(m),p_correct))\r\n\r\n\r\n%%\r\nm = 200;\r\np_correct = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 59, 61, 67, 79, 83, 89, 101, 113, 127, 137, 139, 151, 163, 173, 197, 199];\r\nassert(isequal(prime_as_sum_of_two_consec_primes_pm_1(m),p_correct))\r\n\r\n\r\n%% Test forbidden functions\r\nfiletext = fileread('prime_as_sum_of_two_consec_primes_pm_1.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T07:03:57.000Z","deleted_by":null,"deleted_at":null,"solvers_count":59,"test_suite_updated_at":"2025-07-17T19:19:24.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-17T18:12:11.000Z","updated_at":"2026-03-31T03:46:19.000Z","published_at":"2025-07-17T19:16:00.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSome prime numbers can be written as the sum of two consecutive lower primes plus / minus one :\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"heading\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep_m = p_n + p_{n+1} \\\\pm 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eLike this for example, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e7 = 3 + 5 - 1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e11 = 5 + 7 - 1.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn a vector, list such prime numbers lower than a given -input- positive integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 50   =\u0026gt; p = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 100 =\u0026gt; p = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 59, 61, 67, 79, 83, 89];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em = 200 =\u0026gt; p = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 59, 61, 67, 79, 83, 89, 101, 113, 127, 131, 151, 163, 173, 197, 199];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eFobidden functions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95630\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e I\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":8054,"title":"Stress-Strain Properties - 7","description":"The toughness of a material is technically defined as the plastic strain energy absorbed by the material (the plastic region in the figure below). Practically speaking, it's a measure of how much deformation a material can undergo (or energy it can absorb) before failure.\r\n\r\nWrite a function to calculate the toughness of a material—the absorbed strain energy minus the resilience. This can be accomplished by combining the code written in problem 2 (resilience) and problem 6 (absorbed strain energy). Also, return the fraction of absorbed strain energy that the toughness represents.\r\nPrevious problem: 6 - absorbed strain energy. Next problem: 8 - material properties list.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 499px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 332px 249.5px; transform-origin: 332px 249.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 31.5px; text-align: left; transform-origin: 309px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe toughness of a material is technically defined as the plastic strain energy absorbed by the material (the plastic region in the figure below). Practically speaking, it's a measure of how much deformation a material can undergo (or energy it can absorb) before failure.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 304px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 152px; text-align: center; transform-origin: 309px 152px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline\" src=\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 42px; text-align: left; transform-origin: 309px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eWrite a function to calculate the toughness of a material—the absorbed strain energy minus the resilience. This can be accomplished by combining the code written in\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/8049-stress-strain-properties-2\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eproblem 2\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e (resilience) and\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eproblem 6\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e (absorbed strain energy). Also, return the fraction of absorbed strain energy that the toughness represents.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 10.5px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ePrevious problem: 6 -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eabsorbed strain energy\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. Next problem: 8 -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ematerial properties list\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff)\r\n\r\nT = 1;\r\n\r\nfrac = 0.5;\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile\r\n% in a single direction. Some materials, such as metals, are generally\r\n% isotropic, whereas others, like composites, are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26;\r\nG = 79.3e9; %Pa\r\nE = 200e9; %Pa\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 463e6; %strain-hardening coefficient\r\nT_corr = 12.26e7;\r\nfrac_corr = 0.9987;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\nassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 974e6; %strain-hardening coefficient\r\nT_corr = 11.82e7;\r\nfrac_corr = 0.9751;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\nassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1845e6; %strain-hardening coefficient\r\nT_corr = 3.205e7;\r\nfrac_corr = 0.9067;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\nassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\r\n%% aluminum alloy (6061-T6)\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 325e6; %strain-hardening coefficient\r\nT_corr = 4.279e7;\r\nfrac_corr = 0.9902;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\nassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 304e6; %strain-hardening coefficient\r\nT_corr = 7.340e7;\r\nfrac_corr = 0.9997;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\nassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1870e6; %strain-hardening coefficient\r\nT_corr = 20.05e7;\r\nfrac_corr = 0.9995;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\nassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.1e9; %Pa\r\ndensity = 1.14; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\nT_corr = 3.473e7;\r\nfrac_corr = 0.9697;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\nassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\nT_corr = 0;\r\nfrac_corr = 0;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(isequal(T,T_corr))\r\nassert(isequal(frac,frac_corr))\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\nT_corr = 0;\r\nfrac_corr = 0;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(isequal(T,T_corr))\r\nassert(isequal(frac,frac_corr))\r\n\r\n%%\r\nfor i = 1:30\r\nind = randi(8);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_y = 250e6; %Pa\r\n\t\te_y = 0.00125;\r\n\t\te_u = 0.35;\r\n\t\tsh_exp = 0.14; %strain-hardening exponent\r\n\t\tsh_coeff = 463e6; %strain-hardening coefficient\r\n\t\tT_corr = 12.26e7;\r\n\t\tfrac_corr = 0.9987;\r\n\t\t[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\n\t\tassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\n\t\tassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\tcase 2\r\n\t\tS_y = 830e6; %Pa\r\n\t\te_y = 0.00728;\r\n\t\te_u = 0.14;\r\n\t\tsh_exp = 0.04; %strain-hardening exponent\r\n\t\tsh_coeff = 974e6; %strain-hardening coefficient\r\n\t\tT_corr = 11.82e7;\r\n\t\tfrac_corr = 0.9751;\r\n\t\t[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\n\t\tassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\n\t\tassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\tcase 3\r\n\t\tS_y = 230e6; %Pa\r\n\t\te_y = 0.016;\r\n\t\te_u = 0.016;\r\n\t\tsh_exp = 0; %strain-hardening exponent\r\n\t\tsh_coeff = 0; %strain-hardening coefficient\r\n\t\tT_corr = 0;\r\n\t\tfrac_corr = 0;\r\n\t\t[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\n\t\tassert(isequal(T,T_corr))\r\n\t\tassert(isequal(frac,frac_corr))\r\n\tcase 4\r\n\t\tS_y = 317e6; %Pa\r\n\t\te_y = 0.000685;\r\n\t\te_u = 0.24;\r\n\t\tsh_exp = 0.353; %strain-hardening exponent\r\n\t\tsh_coeff = 1870e6; %strain-hardening coefficient\r\n\t\tT_corr = 20.05e7;\r\n\t\tfrac_corr = 0.9995;\r\n\t\t[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\n\t\tassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\n\t\tassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\tcase 5\r\n\t\tS_y = 70e6; %Pa\r\n\t\te_y = 0.00054;\r\n\t\te_u = 0.48;\r\n\t\tsh_exp = 0.44; %strain-hardening exponent\r\n\t\tsh_coeff = 304e6; %strain-hardening coefficient\r\n\t\tT_corr = 7.340e7;\r\n\t\tfrac_corr = 0.9997;\r\n\tcase 6\r\n\t\tS_y = 1172e6; %Pa\r\n\t\te_y = 0.00563;\r\n\t\te_u = 0.027;\r\n\t\tsh_exp = 0.075; %strain-hardening exponent\r\n\t\tsh_coeff = 1845e6; %strain-hardening coefficient\r\n\t\tT_corr = 3.205e7;\r\n\t\tfrac_corr = 0.9067;\r\n\tcase 7\r\n\t\tS_y = 82e6; %Pa\r\n\t\te_y = 0.0265;\r\n\t\te_u = 0.45;\r\n\t\tsh_exp = 0; %strain-hardening exponent\r\n\t\tsh_coeff = 0; %strain-hardening coefficient\r\n\t\tT_corr = 3.473e7;\r\n\t\tfrac_corr = 0.9697;\r\n\tcase 8\r\n\t\tS_y = 241e6; %Pa\r\n\t\te_y = 0.0035;\r\n\t\te_u = 0.15;\r\n\t\tsh_exp = 0.042; %strain-hardening exponent\r\n\t\tsh_coeff = 325e6; %strain-hardening coefficient\r\n\t\tT_corr = 4.279e7;\r\n\t\tfrac_corr = 0.9902;\r\nend\r\nend % for i = 1:30\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":2,"created_by":26769,"edited_by":26769,"edited_at":"2024-03-27T17:44:32.000Z","deleted_by":null,"deleted_at":null,"solvers_count":88,"test_suite_updated_at":"2021-08-03T17:04:10.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-30T22:03:11.000Z","updated_at":"2026-02-19T09:46:19.000Z","published_at":"2015-03-30T22:03:11.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe toughness of a material is technically defined as the plastic strain energy absorbed by the material (the plastic region in the figure below). Practically speaking, it's a measure of how much deformation a material can undergo (or energy it can absorb) before failure.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"298\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"420\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to calculate the toughness of a material—the absorbed strain energy minus the resilience. This can be accomplished by combining the code written in\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/8049-stress-strain-properties-2\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eproblem 2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e (resilience) and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eproblem 6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e (absorbed strain energy). Also, return the fraction of absorbed strain energy that the toughness represents.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePrevious problem: 6 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eabsorbed strain energy\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. Next problem: 8 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ematerial properties list\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"target\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationshipId\":\"rId1\"}]},{\"partUri\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"contentType\":\"image/net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"content\":\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":8053,"title":"Stress-Strain Properties - 6","description":"The total energy absorbed by a material up to failure in a tensile test is termed the absorbed strain energy. With respect to the figure below, it is the total area of the elastic and plastic regions and can be calculated by integrating the stress-strain curve. As a first approximation, many stress-strain responses can be approximated by:\r\n\r\nwhere K is a strength coefficient, eps_p is the plastic strain, and n is the hardening exponent. Stress as a function of strain can be calculated by creating a strain vector from zero to the ultimate strain and integrating the stress values in that vector.\r\n\r\n(from quora.com)\r\nWrite a function to return the absorbed strain energy for a material provided K and n. If the material does not strain harden, then K and n will be set equal to zero. In these cases, the absorbed strain energy is equal to the resilience (triangular area up to yield point) and any absorbed plastic energy, if applicable, which can be approximated by a rectangle from the yield point to the failure point with those stresses being equal. If the ultimate strain equals the yield strain, that rectangular area is zero.\r\nPrevious problem: 5 - stiffness-to-weight ratio. Next problem: 7 - toughness.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 702px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 332px 351px; transform-origin: 332px 351px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 42px; text-align: left; transform-origin: 309px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe total energy absorbed by a material up to failure in a tensile test is termed the absorbed strain energy. With respect to the figure below, it is the total area of the elastic and plastic regions and can be calculated by integrating the stress-strain curve. As a first approximation, many stress-strain responses can be approximated by:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 29px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 14.5px; text-align: center; transform-origin: 309px 14.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" width=\"73\" height=\"23\" style=\"vertical-align: baseline;width: 73px;height: 23px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 31.5px; text-align: left; transform-origin: 309px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ewhere K is a strength coefficient, eps_p is the plastic strain, and n is the hardening exponent. Stress as a function of strain can be calculated by creating a strain vector from zero to the ultimate strain and integrating the stress values in that vector.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 304px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 152px; text-align: center; transform-origin: 309px 152px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline\" src=\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 10.5px; text-align: center; transform-origin: 309px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e(from quora.com)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 126px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 63px; text-align: left; transform-origin: 309px 63px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eWrite a function to return the absorbed strain energy for a material provided K and n. If the material does not strain harden, then K and n will be set equal to zero. In these cases, the absorbed strain energy is equal to the resilience (triangular area up to yield point) and any absorbed plastic energy, if applicable, which can be approximated by a rectangle from the yield point to the failure point with those stresses being equal. If the ultimate strain equals the yield strain, that rectangular area is zero.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 10.5px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ePrevious problem: 5 -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003estiffness-to-weight ratio\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. Next problem: 7 -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003etoughness\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [ASE] = stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)\r\n\r\nASE = 1;\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile\r\n% in a single direction. Some materials, such as metals are generally\r\n% isotropic, whereas others, like composite are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26;\r\nG = 79.3e9; %Pa\r\nE = 200e9; %Pa\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 463e6; %strain-hardening coefficient\r\nASE_corr = 12.28e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 974e6; %strain-hardening coefficient\r\nASE_corr = 12.12e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1845e6; %strain-hardening coefficient\r\nASE_corr = 3.535e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%% aluminum alloy (6061-T6)%^\u0026\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 325e6; %strain-hardening coefficient\r\nASE_corr = 4.321e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 304e6; %strain-hardening coefficient\r\nASE_corr = 7.342e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1870e6; %strain-hardening coefficient\r\nASE_corr = 20.06e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.1e9; %Pa\r\ndensity = 1.14; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\nASE_corr = 3.581e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\nASE_corr = 0.184e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\nASE_corr = 0.06e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_y = 250e6; %Pa\r\n\t\te_y = 0.00125;\r\n\t\te_u = 0.35;\r\n\t\tsh_exp = 0.14; %strain-hardening exponent\r\n\t\tsh_coeff = 463e6; %strain-hardening coefficient\r\n\t\tASE_corr = 12.28e7;\r\n\tcase 2\r\n\t\tS_y = 82e6; %Pa\r\n\t\te_y = 0.0265;\r\n\t\te_u = 0.45;\r\n\t\tsh_exp = 0; %strain-hardening exponent\r\n\t\tsh_coeff = 0; %strain-hardening coefficient\r\n\t\tASE_corr = 3.581e7;\r\n\tcase 3\r\n\t\tS_y = 241e6; %Pa\r\n\t\te_y = 0.0035;\r\n\t\te_u = 0.15;\r\n\t\tsh_exp = 0.042; %strain-hardening exponent\r\n\t\tsh_coeff = 325e6; %strain-hardening coefficient\r\n\t\tASE_corr = 4.321e7;\r\n\tcase 4\r\n\t\tS_y = 317e6; %Pa\r\n\t\te_y = 0.000685;\r\n\t\te_u = 0.24;\r\n\t\tsh_exp = 0.353; %strain-hardening exponent\r\n\t\tsh_coeff = 1870e6; %strain-hardening coefficient\r\n\t\tASE_corr = 20.06e7;\r\nend\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_y = 830e6; %Pa\r\n\t\te_y = 0.00728;\r\n\t\te_u = 0.14;\r\n\t\tsh_exp = 0.04; %strain-hardening exponent\r\n\t\tsh_coeff = 974e6; %strain-hardening coefficient\r\n\t\tASE_corr = 12.12e7;\r\n\tcase 2\r\n\t\tS_y = 241e6; %Pa\r\n\t\te_y = 0.0035;\r\n\t\te_u = 0.15;\r\n\t\tsh_exp = 0.042; %strain-hardening exponent\r\n\t\tsh_coeff = 325e6; %strain-hardening coefficient\r\n\t\tASE_corr = 4.321e7;\r\n\tcase 3\r\n\t\tS_y = 250e6; %Pa\r\n\t\te_y = 0.00125;\r\n\t\te_u = 0.35;\r\n\t\tsh_exp = 0.14; %strain-hardening exponent\r\n\t\tsh_coeff = 463e6; %strain-hardening coefficient\r\n\t\tASE_corr = 12.28e7;\r\n\tcase 4\r\n\t\tS_y = 70e6; %Pa\r\n\t\te_y = 0.00054;\r\n\t\te_u = 0.48;\r\n\t\tsh_exp = 0.44; %strain-hardening exponent\r\n\t\tsh_coeff = 304e6; %strain-hardening coefficient\r\n\t\tASE_corr = 7.342e7;\r\nend\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_y = 1200e6; %Pa\r\n\t\te_y = 0.001;\r\n\t\te_u = 0.001;\r\n\t\tsh_exp = 0; %strain-hardening exponent\r\n\t\tsh_coeff = 0; %strain-hardening coefficient\r\n\t\tASE_corr = 0.06e7;\r\n\tcase 2\r\n\t\tS_y = 250e6; %Pa\r\n\t\te_y = 0.00125;\r\n\t\te_u = 0.35;\r\n\t\tsh_exp = 0.14; %strain-hardening exponent\r\n\t\tsh_coeff = 463e6; %strain-hardening coefficient\r\n\t\tASE_corr = 12.28e7;\r\n\tcase 3\r\n\t\tS_y = 230e6; %Pa\r\n\t\te_y = 0.016;\r\n\t\te_u = 0.016;\r\n\t\tsh_exp = 0; %strain-hardening exponent\r\n\t\tsh_coeff = 0; %strain-hardening coefficient\r\n\t\tASE_corr = 0.184e7;\r\n\tcase 4\r\n\t\tS_y = 1172e6; %Pa\r\n\t\te_y = 0.00563;\r\n\t\te_u = 0.027;\r\n\t\tsh_exp = 0.075; %strain-hardening exponent\r\n\t\tsh_coeff = 1845e6; %strain-hardening coefficient\r\n\t\tASE_corr = 3.535e7;\r\nend\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":7,"created_by":26769,"edited_by":26769,"edited_at":"2024-03-27T17:39:31.000Z","deleted_by":null,"deleted_at":null,"solvers_count":92,"test_suite_updated_at":"2015-03-30T21:25:46.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-30T20:58:35.000Z","updated_at":"2026-02-19T09:44:40.000Z","published_at":"2015-03-30T21:25:46.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe total energy absorbed by a material up to failure in a tensile test is termed the absorbed strain energy. With respect to the figure below, it is the total area of the elastic and plastic regions and can be calculated by integrating the stress-strain curve. As a first approximation, many stress-strain responses can be approximated by:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"23\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"73\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere K is a strength coefficient, eps_p is the plastic strain, and n is the hardening exponent. Stress as a function of strain can be calculated by creating a strain vector from zero to the ultimate strain and integrating the stress values in that vector.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"298\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"420\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e(from quora.com)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to return the absorbed strain energy for a material provided K and n. If the material does not strain harden, then K and n will be set equal to zero. In these cases, the absorbed strain energy is equal to the resilience (triangular area up to yield point) and any absorbed plastic energy, if applicable, which can be approximated by a rectangle from the yield point to the failure point with those stresses being equal. If the ultimate strain equals the yield strain, that rectangular area is zero.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePrevious problem: 5 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003estiffness-to-weight ratio\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. Next problem: 7 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003etoughness\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"target\":\"/media/image1.png\",\"relationshipId\":\"rId1\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"target\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationshipId\":\"rId2\"}]},{\"partUri\":\"/media/image1.png\",\"contentType\":\"image/png\",\"content\":\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEkAAAAXBAMAAAC12X9oAAAAMFBMVEX///8WFhZQUFBAQEAwMDBiYmLm5uZ0dHTMzMyKiooMDAyenp4EBAQiIiK2trYAAAARmbiRAAAAAXRSTlMAQObYZgAAAU9JREFUeNpjZiAAUsUCn31gwipV+8ebgfv/PRBTO7b7AgN2VRP/bmX4fVMJxCzz5tuAQ9X3vwwM9zXATOYHXxpwqOJYxcB5GsL81fD/KA4n8xswRECZLA1M0QzMDHw3JjQ4LkBV9YJR9d8FEGPrnccH/l8G2rhmZ8NzB6CAoSAIgOUYCu49CgDRDw+egmhj+8fAG4Buo9dsfpBGtpNQPtM/IwaGC+iqGp5+2Qak/l2HqeLewPC3AN2Lug2sf4A0d7FxA0SkH4gSGFDdxf6DgUEG5NUJMI8C8eJzINZ5MN8ARPw9wcDQ/LGA4S/cdMYGTgxn8QCdzi8ANNMAKsDM9sU8F00Rs8SU/9dczyVN+BtrW8q7nH35BEZCKWeCZCoTjpSDBArE/05gIKiKW+D7BAaCNjLJeX1bQEgRA6+6BQNhwAM2kJCqq8SoYvu5AUgCAMD/X3IO0icRAAAAAElFTkSuQmCC\",\"relationship\":null},{\"partUri\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"contentType\":\"image/net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"content\":\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60953,"title":"Chek the Delta =  p' - p = 6k gap theorem about arithmetic progressions in the prime number set","description":"Context\r\n \r\nIn the prime numbers set there are some arithmetic progressions (sequences of three or more consecutive prime numbers (p, p’, p’’) equally spaced one to the others by an even number  ).\r\n \r\nOne theorem, which can actually easily be proven from , is that above the sole and unique triplet (3, 5, 7) -with a gap of  then-  all the following progressions are such that \r\n \r\nProblem statement\r\n\r\nFor a given interval [i1, i2], i1 \u003e 7 and i2 \u003e 7 find p the first corresponding arithmetic progression (3 or more consecutive consecutive primes equally spaced) in this interval and check the conjecture equation simply by calculating the integer ratio . \r\n\r\nExamples\r\n                \r\nFor [i1, i2] = [8, 68], p = [47, 53, 59] and k = [1, 1], since this is the first arithmetic progression above 8 and with  here; \r\n \r\nFor [i1, i2] = [180, 228], p = [199, 211, 223], and k = [2, 2], since this is the first arithmetic progression above 180 and with  here;\r\n\r\nFor [i1, i2] = [240, 272], p = [251, 257, 263, 269], and k = [1, 1, 1], since this is the first arithmetic progression above 140 and with  here; \r\n\r\nFor [i1, i2] = [180, 272], p = [199, 211, 223], and k = [2, 2], since this is the first arithmetic progression above 180 and with  here;\r\n\r\nTip\r\n \r\nFirst maybe, train yourself to find the first and last indices of zeros of a first block of consecutive zeros in a vector of integers, eg for u = [1 0 0 0 1 1 0 0 1], j1 = 2 and j2 = 4.\r\n\r\nForbidden functions\r\n \r\n \r\nregexp\r\n \r\nstr2num\r\n \r\nassignin\r\n\r\necho\r\n \r\n \r\nSee also\r\n \r\nProblem 60940. Find the first occurence of a given gap between two consecutive prime numbers\r\nPrime numbers properties I","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 1469.2px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 734.6px; transform-origin: 408px 734.6px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.0583px 8px; transform-origin: 26.0583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eContext\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 248.55px 8px; transform-origin: 248.55px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the prime numbers set there are some arithmetic progressions (sequences of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.45px 8px; transform-origin: 110.45px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003ethree or more consecutive prime numbers\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.825px 8px; transform-origin: 30.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e(p, p’, p’’)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 166.875px 8px; transform-origin: 166.875px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e equally spaced one to the others by an even number \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"134.5\" height=\"18\" style=\"width: 134.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.21667px 8px; transform-origin: 6.21667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e ).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 172.317px 8px; transform-origin: 172.317px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eOne theorem, which can actually easily be proven from \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"236.5\" height=\"19\" style=\"width: 236.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 85.5667px 8px; transform-origin: 85.5667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, is that above the sole and unique triplet \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.1083px 8px; transform-origin: 24.1083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e(3, 5, 7)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 45.8917px 8px; transform-origin: 45.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e -with a gap of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"40\" height=\"18\" style=\"width: 40px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.8917px 8px; transform-origin: 17.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e then-\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 134.208px 8px; transform-origin: 134.208px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eall the following progressions are such that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"97\" height=\"19\" style=\"width: 97px; height: 19px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 31.5px; text-align: left; transform-origin: 385px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 61.4583px 8px; transform-origin: 61.4583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor a given interval \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 75.825px 8px; transform-origin: 75.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e[i1, i2], i1 \u0026gt; 7 and i2 \u0026gt; 7\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.1667px 8px; transform-origin: 15.1667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e find \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4.275px 8px; transform-origin: 4.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 217.05px 8px; transform-origin: 217.05px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e the first corresponding arithmetic progression (3 or more consecutive consecutive primes equally spaced) in this interval and check the conjecture equation simply by calculating the integer ratio \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"57\" height=\"18.5\" style=\"width: 57px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 31.0667px 8px; transform-origin: 31.0667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 40.8667px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 20.4333px; transform-origin: 392px 20.4333px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 12.4417px 8px; transform-origin: 12.4417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 48.4167px 8px; transform-origin: 48.4167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e[i1, i2] = [8, 68]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 48.0333px 8px; transform-origin: 48.0333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [47, 53, 59]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 28.2px 8px; transform-origin: 28.2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ek = [1, 1]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 158.692px 8px; transform-origin: 158.692px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, since this is the first arithmetic progression above \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e8\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 29.95px 8px; transform-origin: 29.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"90\" height=\"18\" style=\"width: 90px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 19.8333px 8px; transform-origin: 19.8333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e here; \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 40.8667px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 20.4333px; transform-origin: 392px 20.4333px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 12.4417px 8px; transform-origin: 12.4417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 60.0917px 8px; transform-origin: 60.0917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e[i1, i2] = [180, 228]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 59.1917px 8px; transform-origin: 59.1917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [199, 211, 223]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 32.0833px 8px; transform-origin: 32.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ek = [2, 2], \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 154.808px 8px; transform-origin: 154.808px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince this is the first arithmetic progression above \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e180\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"97.5\" height=\"18\" style=\"width: 97.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 17.8917px 8px; transform-origin: 17.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e here;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 40.8667px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 20.4333px; transform-origin: 392px 20.4333px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 12.4417px 8px; transform-origin: 12.4417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 60.0917px 8px; transform-origin: 60.0917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e[i1, i2] = [240, 272]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 75.2667px 8px; transform-origin: 75.2667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [251, 257, 263, 269]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 39.8583px 8px; transform-origin: 39.8583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ek = [1, 1, 1], \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 154.808px 8px; transform-origin: 154.808px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince this is the first arithmetic progression above \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e140\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 29.95px 8px; transform-origin: 29.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"90\" height=\"18\" style=\"width: 90px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 19.8333px 8px; transform-origin: 19.8333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e here; \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 40.8667px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 20.4333px; transform-origin: 392px 20.4333px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 12.4417px 8px; transform-origin: 12.4417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 60.0917px 8px; transform-origin: 60.0917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e[i1, i2] = [180, 272]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 59.1917px 8px; transform-origin: 59.1917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [199, 211, 223]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 32.0833px 8px; transform-origin: 32.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ek = [2, 2], \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 154.808px 8px; transform-origin: 154.808px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince this is the first arithmetic progression above \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e180\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"97.5\" height=\"18\" style=\"width: 97.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 17.8917px 8px; transform-origin: 17.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e here;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.3667px 8px; transform-origin: 10.3667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eTip\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 358.192px 8px; transform-origin: 358.192px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFirst maybe, train yourself to find the first and last indices of zeros of a first block of consecutive zeros in a vector of integers, eg for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 122.325px 8px; transform-origin: 122.325px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eu = [1 0 0 0 1 1 0 0 1], j1 = 2 and j2 = 4.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 67.6417px 8px; transform-origin: 67.6417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eForbidden functions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/problems/60940-find-the-first-occurence-of-a-given-gap-between-two-consecutive-prime-numbers\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eProblem 60940. Find the first occurence of a given gap between two consecutive prime numbers\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties I\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [p, k] = check_the_6k_delta_theorem(i1, i2)\r\n  \r\n    u = i1;\r\n    k = i2;\r\n\r\nend","test_suite":"%%\r\ni1 = 8;\r\ni2 = 68;\r\np_correct = [47, 53, 59];\r\nk_correct = [1, 1];\r\n[p,k] = check_the_6k_delta_theorem(i1,i2);\r\nassert(isequal(p,p_correct) \u0026 isequal(k,k_correct))\r\n\r\n\r\n%%\r\ni1 = 180;\r\ni2 = 228;\r\np_correct = [199, 211, 223];\r\nk_correct = [2, 2];\r\n[p,k] = check_the_6k_delta_theorem(i1,i2);\r\nassert(isequal(p,p_correct) \u0026 isequal(k,k_correct))\r\n\r\n\r\n%%\r\ni1 = 240;\r\ni2 = 272;\r\np_correct = [251, 257, 263, 269];\r\nk_correct = [1, 1, 1];\r\n[p,k] = check_the_6k_delta_theorem(i1,i2);\r\nassert(isequal(p,p_correct) \u0026 isequal(k,k_correct))\r\n\r\n\r\n%%\r\ni1 = 180;\r\ni2 = 272;\r\np_correct = [199, 211, 223];\r\nk_correct = [2, 2];\r\n[p,k] = check_the_6k_delta_theorem(i1,i2);\r\nassert(isequal(p,p_correct) \u0026 isequal(k,k_correct))\r\n\r\n\r\n%% Test forbidden functions\r\nfiletext = fileread('check_the_6k_delta_theorem.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":4,"comments_count":3,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T07:04:42.000Z","deleted_by":null,"deleted_at":null,"solvers_count":34,"test_suite_updated_at":"2025-07-11T05:36:42.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-05T06:40:29.000Z","updated_at":"2026-03-06T14:30:46.000Z","published_at":"2025-07-10T12:03:38.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eContext\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the prime numbers set there are some arithmetic progressions (sequences of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ethree or more consecutive prime numbers\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e(p, p’, p’’)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e equally spaced one to the others by an even number \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mathbf{\\\\Delta = p' - p = p'' - p'}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e ).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOne theorem, which can actually easily be proven from \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\forall p \\\\in \\\\mathbb{P}, p\u0026gt; 3 \\\\Rightarrow \\\\exists n \\\\in \\\\mathbb{N}^*, p = 6n \\\\pm 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, is that above the sole and unique triplet \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e(3, 5, 7)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e -with a gap of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e then-\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eall the following progressions are such that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mathbf{\\\\Delta = 6k}, k \\\\in \\\\mathbb{N}^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor a given interval \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[i1, i2], i1 \u0026gt; 7 and i2 \u0026gt; 7\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e find \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e the first corresponding arithmetic progression (3 or more consecutive consecutive primes equally spaced) in this interval and check the conjecture equation simply by calculating the integer ratio \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mathbf{k = \\\\Delta / 6}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[i1, i2] = [8, 68]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [47, 53, 59]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek = [1, 1]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, since this is the first arithmetic progression above \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e8\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 6 = \\\\mathbf{1} \\\\times 6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e here; \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[i1, i2] = [180, 228]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [199, 211, 223]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek = [2, 2], \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince this is the first arithmetic progression above \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e180\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 12 = \\\\mathbf{2} \\\\times 6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e here;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[i1, i2] = [240, 272]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [251, 257, 263, 269]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek = [1, 1, 1], \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince this is the first arithmetic progression above \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e140\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 6 = \\\\mathbf{1} \\\\times 6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e here; \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[i1, i2] = [180, 272]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [199, 211, 223]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek = [2, 2], \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince this is the first arithmetic progression above \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e180\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 12 = \\\\mathbf{2} \\\\times 6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e here;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eTip\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFirst maybe, train yourself to find the first and last indices of zeros of a first block of consecutive zeros in a vector of integers, eg for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eu = [1 0 0 0 1 1 0 0 1], j1 = 2 and j2 = 4.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc 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