Partitions with equal products and elliptic curves

TL;DR

This paper explicitly describes the Mordell-Weil group of a family of elliptic curves related to partitions of integers with equal products, revealing positive rank and infinitely many such partitions.

Contribution

It provides an explicit description of the Mordell-Weil group for these elliptic curves and determines their torsion subgroup, establishing the existence of positive rank and infinitely many partitions.

Findings

Mordell-Weil group explicitly described

Torsion subgroup determined

Infinitely many partitions with equal product and sum

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 $\displaystyle E_{(M,N)}:y^2-Mxy-Ny=x^3$ over $\Q$. In particular we determine the torsion subgroup of $E_{(M,N)}(\Q)$ 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 $E_{(M,N)}(\Q)$ has rank at least $n$.