Partitions into Triples with Equal Products and Families of Elliptic Curves

TL;DR

This paper explores the connection between partitions of integers into triples with equal products and families of elliptic curves, achieving high-rank examples through parametric constructions and computational searches.

Contribution

It establishes a novel link between integer partitions with equal sums and products and elliptic curve families, providing explicit high-rank examples.

Findings

Families of elliptic curves with rank ≥ 5, 6, and 8 for different partition sizes.

Construction of specific elliptic curves with rank ≥ 11 through computer search.

Discovery of two elliptic curves with rank 14.

Abstract

Let $S_l(M,N)$ denote a set of $\ell$ triples of positive integers having the same sum $M$ and the same product $N$. For each $2\leq\ell\leq 4$ we establish a connection between a subset of $S_l(M,N)$ with (integral) parametric elements and a family of elliptic curves. When $\ell=2$ and $3$, we use certain known subsets of $S_l(M,N)$ with parametric elements and respectively find families of elliptic curves of generic rank $\geq 5$ and $\geq 6$, while for $\ell=4$ we first obtain a subset of $S_l(M,N)$ with parametric elements, then construct a family of elliptic curves of generic rank $\geq 8$. Finally, we perform a computer search within these families to find specific curves with rank $\geq 11$ and in particular we found two curves of rank $14$.