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arXiv · 2408.13867

Partitions into Triples with Equal Products and Families of Elliptic Curves

Ahmed El Amine Youmbai·Arman Shamsi Zargar·Maksym Voznyy
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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 $14$.

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Ahmed El Amine Youmbai, Arman Shamsi Zargar, Maksym Voznyy. 2024-08-25. Partitions into Triples with Equal Products and Families of Elliptic Curves. https://arxiv.org/abs/2408.13867

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