Glasnik Matematicki, Vol. 60, No. 1 (2025), 59-72. \( \)

PARTITIONS INTO TRIPLES WITH EQUAL PRODUCTS AND FAMILIES OF ELLIPTIC CURVES

Ahmed El Amine Youmbai, Arman Shamsi Zargar and Maksym Voznyy

LABTHOP Laboratory, Mathematics Department, Faculty of Exact Sciences, University of El Oued, PO Box 789, 39000 Echott El Oued, Algeria
e-mail:youmbai-amine@univ-eloued.dz

Department of Mathematics and Applications, Faculty of Mathematical Sciences, University of Mohaghegh Ardabili, Ardabil, Iran
e-mail:zargar@uma.ac.ir

Department of Technology, Stephen Leacock CI, Toronto District School Board, Toronto, Canada
e-mail:maksym.voznyy@tdsb.on.ca

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 \(14\).

2020 Mathematics Subject Classification.   14H52, 11D25, 11D09, 05A17

Key words and phrases.   Triples of integers, equal sum of integers, equal products, Diophantine equation, partition, parametric solution, elliptic curve

Full text (PDF) (access from subscribing institutions only)

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

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