Partitions into Triples with Equal Products and Families of 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 $2\le \ell \le 4$ 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 $\ge 5$ and $\ge 6$, 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 $\ge 8$. Finally, we perform a computer search within these families to find specific curves with rank $\ge 11$ and in particular we found two curves of rank $14$.

12 pages, 4 tables