Some new families of positive-rank elliptic curves arising from Pythagorean triples

Mehdi Baghalaghdam and Farzali Izadi
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 24, 2018, Number 3, Pages 27–36
DOI: 10.7546/nntdm.2018.24.3.27-36
Full paper (PDF, 194 Kb)

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Authors and affiliations

Mehdi Baghalaghdam
Department of Mathematics, Faculty of Science,
Azarbaijan Shahid Madani University, Tabriz 53751-71379, Iran

Farzali Izadi
Department of Mathematics, Faculty of Science,
Urmia University, Urmia 165-57153, Iran

Abstract

In the present paper, we introduce some new families of elliptic curves with positive rank arising from Pythagorean triples. We study elliptic curves of the form y² = x³ – A²x + B², where A, B ∈ {a, b, c} are two different numbers and (a, b, c) is a rational Pythagorean triple. First of all, we prove that if (a, b, c) is a primitive Pythagorean triple (PPT), then the rank of each family is positive. Furthermore, we construct subfamilies of rank at least 3 in each family but one with rank at least 2, and obtain elliptic curves of high rank in each family. Finally, we consider two other new families of elliptic curves of the forms y² = x(x – a²)(x + c²) and y² = x(x – b²)(x + c²), and prove that if (a, b, c) is a PPT, then the rank of each family is positive.

Keywords

• Elliptic curves
• Rank
• Pythagorean triples

2010 Mathematics Subject Classification

• 11G05
• 14H52
• 14G05

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