Problem 51471. Find integer solutions to an equation with a sum of binomial coefficients

Created by ChrisR

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<div 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; "><div style="block-size: 145.25px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 72.625px; transform-origin: 407px 72.625px; 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: 384px 10.5px; text-align: left; transform-origin: 384px 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: 115.125px 7.91667px; transform-origin: 115.125px 7.91667px; unicode-bidi: normal; ">Theorem 1 of chapter 27 of the book <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: 70.425px 7.91667px; transform-origin: 70.425px 7.91667px; unicode-bidi: normal; ">Diophantine Equations<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: 198.45px 7.91667px; transform-origin: 198.45px 7.91667px; unicode-bidi: normal; "> by Louis J. Mordell shows that the equation <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 7.91667px; transform-origin: 0px 7.91667px; unicode-bidi: normal; "> <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 7.91667px; transform-origin: 0px 7.91667px; unicode-bidi: normal; "> <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 7.91667px; transform-origin: 0px 7.91667px; unicode-bidi: normal; "> <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 7.91667px; transform-origin: 0px 7.91667px; unicode-bidi: normal; "> <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 7.91667px; transform-origin: 0px 7.91667px; unicode-bidi: normal; "> <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 7.91667px; transform-origin: 0px 7.91667px; unicode-bidi: normal; "> <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 7.91667px; transform-origin: 0px 7.91667px; unicode-bidi: normal; "> <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 7.91667px; transform-origin: 0px 7.91667px; unicode-bidi: normal; "> <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 7.91667px; transform-origin: 0px 7.91667px; unicode-bidi: normal; "> <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 7.91667px; transform-origin: 0px 7.91667px; unicode-bidi: normal; "> <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 7.91667px; transform-origin: 0px 7.91667px; unicode-bidi: normal; "> <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 7.91667px; transform-origin: 0px 7.91667px; unicode-bidi: normal; "> <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 7.91667px; transform-origin: 0px 7.91667px; unicode-bidi: normal; "> <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 7.91667px; transform-origin: 0px 7.91667px; unicode-bidi: normal; "> <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 7.91667px; transform-origin: 0px 7.91667px; unicode-bidi: normal; "> <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 7.91667px; transform-origin: 0px 7.91667px; unicode-bidi: normal; "> <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 7.91667px; transform-origin: 0px 7.91667px; unicode-bidi: normal; "> <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 7.91667px; transform-origin: 0px 7.91667px; unicode-bidi: normal; "> </div><div style="block-size: 34px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-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 17px; text-align: left; transform-origin: 384px 17px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><img src="data:image/png;base64,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" alt="y^2 = (x choose 0) + (x choose 1) + (x choose 2) + (x choose 3)" style="width: 186.5px; height: 34px;" width="186.5" height="34"></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: 384px 10.5px; text-align: left; transform-origin: 384px 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: 137.7px 7.91667px; transform-origin: 137.7px 7.91667px; unicode-bidi: normal; ">has a finite number of real, integer solutions <img src="data:image/png;base64,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" alt="(x,y)" style="width: 36px; height: 19px;" width="36" height="19"><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: 57.55px 7.91667px; transform-origin: 57.55px 7.91667px; unicode-bidi: normal; ">. One of them has <img src="data:image/png;base64,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" alt="x = 7" style="width: 36.5px; height: 18px;" width="36.5" height="18"><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: 30.3417px 7.91667px; transform-origin: 30.3417px 7.91667px; unicode-bidi: normal; "> because <img src="data:image/png;base64,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" alt="y^2 = 1 + 7 + 21 + 35 + 64 = 8^2" style="width: 195px; height: 19.5px;" width="195" height="19.5"><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: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; ">. </div><div style="block-size: 42.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-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 21.125px; text-align: left; transform-origin: 384px 21.125px; 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: 107.217px 7.91667px; transform-origin: 107.217px 7.91667px; unicode-bidi: normal; ">Write a function that takes a value <img src="data:image/png;base64,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" alt="x = xmax" style="width: 52.5px; height: 20px;" width="52.5" height="20"><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 7.91667px; transform-origin: 0px 7.91667px; unicode-bidi: normal; "><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: 118.242px 7.91667px; transform-origin: 118.242px 7.91667px; unicode-bidi: normal; "> and returns a two-column matrix with <img src="data:image/png;base64,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" alt="(x, y)" style="width: 36px; height: 19px;" width="36" height="19"><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: 29.1667px 7.91667px; transform-origin: 29.1667px 7.91667px; unicode-bidi: normal; "> pairs for <img src="data:image/png;base64,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" alt="x <= xmax" style="width: 52.5px; height: 20px;" width="52.5" height="20"><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: 54.8333px 7.91667px; transform-origin: 54.8333px 7.91667px; unicode-bidi: normal; ">. Sort the pairs in order of increasing x<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: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; ">.</div></div></div>

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Last Solution submitted on Aug 11, 2021

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goc3 on 20 Apr 2021

Superb job of not giving away answers in the test suite.

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Solution 5391667

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2 Comments

Tim on 19 Apr 2021

The solutions for x<-1 would require y^2<0.

ChrisR on 19 Apr 2021

William, expand the right side.

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binomial diophantine

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Problem 51471. Find integer solutions to an equation with a sum of binomial coefficients

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Problem 51471. Find integer solutions to an equation with a sum of binomial coefficients

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