Partitions into Triples with Equal Products and Families of Elliptic Curves

TL;DR

This work studies partitions of triples with equal sum $M$ and product $N$, formalized via ${\mathcal{S}}_{\ell}^{(1)}(M,N)$, and builds parametric subsets connected to families of elliptic curves. For $(\ell,n)=(2,1),(3,1),(4,1)$, it constructs three elliptic-curve families ${\mathcal{E}}_{\mathcal{A}}, {\mathcal{E}}_{\mathcal{B}}, {\mathcal{E}}_{\mathcal{C}}$ with generic rank lower bounds $\ge 5$, $\ge 6$, and $\ge 8$ respectively, using Néron–Silverman specialization to certify independence of rational points. Computer searches within these families yield explicit high-rank curves, including two curves of rank $14$ and several instances with rank $\ge 11$, illustrating the practical impact of the method. Heuristics based on Mestre–Nagao sums guide the selection of promising parameter values. The work raises natural questions about extending the approach to more than four partitions and exploring further connections between symmetric Diophantine systems and high-rank elliptic curves.

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 $14$.

Theorems & Definitions (14)

• Theorem 2.1
• Remark 2.2
• Theorem 2.3
• proof
• Remark 2.4
• Remark 2.5
• Theorem 2.6
• proof
• Remark 2.7
• Proposition 2.8