Problem 55330. Convert Periodic Continued Fraction to Fractional Radical Representation

Created by David Hill

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<div style = "text-align: start; line-height: 20.44px; 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 solid rgb(0, 0, 0); white-space: normal; "><div style="block-size: 63.5px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 31.75px; transform-origin: 407px 31.75px; vertical-align: baseline; "><div style="font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.75px; text-align: left; transform-origin: 384px 31.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-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; "><span style="">Every periodic continued fraction can be prepresented by a number of the form </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="76.5" height="21" style="width: 76.5px; height: 21px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-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; "><span style=""> where p, q, and d are all integers with d&gt;0, </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="36.5" height="18" style="width: 36.5px; height: 18px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-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; "><span style="">, and d not a perfect square. Given the cointued fraction sequence, both the beginning sequence and cyclic part of the sequence [front, cyclic], output the unique p, q, and d (in reduced form). p and q can both be negative.</span></span></div></div></div>

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Last Solution submitted on Nov 30, 2022

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Tim on 1 Sep 2022

Tests 8 and 9 have Z(1) three times instead of, presumably, Z(1), Z(2), and Z(3) (not that it's going to help me any).

David Hill on 1 Sep 2022

Thanks, that was not my intent. I corrected the test suite.

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Problem 55330. Convert Periodic Continued Fraction to Fractional Radical Representation

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Problem 55330. Convert Periodic Continued Fraction to Fractional Radical Representation

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