An unusual cubic representation problem

• Published in 2014
• Added on 2017-08-07

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).

Links

• http://ami.ektf.hu/uploads/papers/finalpdf/AMI_43_from29to41.pdf

Other information

BibTeX entry

@article{AnUnusualCubicRepresentationProblem,
	key = {AnUnusualCubicRepresentationProblem},
	type = {article},
	title = {An unusual cubic representation problem},
	author = {Andrew Bremner and Allan Macleod},
	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).},
	comment = {},
	date_added = {2017-08-07},
	date_published = {2014-10-09},
	urls = {http://ami.ektf.hu/uploads/papers/finalpdf/AMI{\_}43{\_}from29to41.pdf},
	collections = {Puzzles},
	url = {http://ami.ektf.hu/uploads/papers/finalpdf/AMI{\_}43{\_}from29to41.pdf},
	urldate = {2017-08-07},
	year = 2014
}