Interesting Esoterica

An unusual cubic representation problem

Article by Andrew Bremner and Allan Macleod
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For a non-zero integer \(N\), we consider the problem of finding \(3\) integers \( (a, b, c) \) such that \[ N = \frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}. \] We show that the existence of solutions is related to points of infinite order on a family of elliptic curves. We discuss strictly positive solutions and prove the surprising fact that such solutions do not exist for \(N\) odd, even though there may exist solutions with one of \(a, b, c\) negative. We also show that, where a strictly positive solution does exist, it can be of enormous size (trillions of digits, even in the range we consider).

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BibTeX entry

@article{AnUnusualCubicRepresentationProblem,
	title = {An unusual cubic representation problem},
	abstract = {For a non-zero integer \(N\), we consider the problem of finding \(3\) integers
\( (a, b, c) \) such that

\[ N = \frac{\{}a{\}}{\{}b+c{\}} + \frac{\{}b{\}}{\{}c+a{\}} + \frac{\{}c{\}}{\{}a+b{\}}. \]

We show that the existence of solutions is related to points of infinite order on a family of elliptic curves. We discuss strictly positive solutions and prove the surprising fact that such solutions do not exist for \(N\) odd, even though there may exist solutions with one of \(a, b, c\) negative. We also show that, where a strictly positive solution does exist, it can be of enormous size (trillions of digits, even in the range we consider).},
	url = {http://ami.ektf.hu/uploads/papers/finalpdf/AMI{\_}43{\_}from29to41.pdf},
	author = {Andrew Bremner and Allan Macleod},
	comment = {},
	urldate = {2017-08-07},
	collections = {puzzles}
}