Partitions with equal products and elliptic curves

Abstract

Let $a$, $b$, $c$ be distinct positive integers. Set $M = a+b+c$ and $N = abc$. We give an explicit description of the Mordell--Weil group of the elliptic curve $E_{(M, N)}\colon y^{2}-Mxy-Ny = x^{3}$ over $\mathbb{Q}$. In particular we determine the torsion subgroup of $E_{(M, N)}(\mathbb{Q})$ and show that its rank is positive. Furthermore there are infinitely many positive integers $M$ that can be written in $n$ different ways, $n\in\{2, 3\}$, as the sum of three distinct positive integers with the same product $N$ and $E_{(M, N)}(\mathbb{Q})$ has rank at least $n$.