Original scientific paper

https://doi.org/10.3336/gm.60.1.04

Partitions into triples with equal products and families of elliptic curves

Ahmed El Amine Youmbai ; LABTHOP Laboratory, Mathematics Department, Faculty of Exact Sciences, University of El Oued, PO Box 789, 39000 Echott El Oued, Algeria
Arman Shamsi Zargar ; Department of Mathematics and Applications, Faculty of Mathematical Sciences, University of Mohaghegh Ardabili, Ardabil, Iran
Maksym Voznyy orcid.org/0000-0001-5649-8416 ; Department of Technology, Stephen Leacock CI, Toronto District School Board, Toronto, Canada

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Abstract

Let \({\mathcal{S}_{\ell}}(M,N)\) denote a set of \(\ell\) (distinct) 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 \({\mathcal{S}_{\ell}}(M,N)\) with (integral) parametric elements and a family of elliptic curves. When \(\ell=2\) and \(3\), we use certain known subsets of \({\mathcal{S}_{\ell}}(M,N)\) 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 \({\mathcal{S}_{\ell}}(M,N)\) 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 \(14\).

Keywords

Triples of integers, equal sum of integers, equal products, Diophantine equation, partition, parametric solution, elliptic curve

Hrčak ID:

332477

URI

https://hrcak.srce.hr/332477

Publication date:

15.8.2026.

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