Problem 57477. Solve an equation involving primes and fractions

Created by ChrisR

Solve Later

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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: 146px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 343px 73px; transform-origin: 343px 73px; vertical-align: baseline; "><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 320px 10.5px; text-align: left; transform-origin: 320px 10.5px; 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="">Write a function to find pairs of primes </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">p</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 </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">q</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=""> satisfying the equation</span></span></div><div style="block-size: 35px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 320px 17.5px; text-align: left; transform-origin: 320px 17.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-15px"><img 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width="136" height="35" alt="p/(p+1) + (q+1)/q = 2n/(n+2)" style="width: 136px; height: 35px;"></span></div><div 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: 320px 21px; text-align: left; transform-origin: 320px 21px; 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="">where </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</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=""> is an integer. The function should take a number </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">x</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=""> as input and produce the triples </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">p</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="">, </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">q</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 </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</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=""> such that </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="38" height="18" alt="p <= x" style="width: 38px; 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="">. If there are no solutions, the function should return three empty vectors.</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 320px 10.5px; text-align: left; transform-origin: 320px 10.5px; 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="">This problem is adapted from one in the </span></span><a target='_blank' href = "https://www.egmo2012.org.uk/"><span style=""><span style="text-decoration: underline; text-decoration-line: underline; ">2012 European Girls’ Math Olympiad</span></span></a><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="">. </span></span></div></div></div>

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Problem 57477. Solve an equation involving primes and fractions

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