# An unusual cubic representation problem

• Published in 2014
In the collection
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).

### BibTeX entry

@article{AnUnusualCubicRepresentationProblem,
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).},
}