New methods for obtaining new families of congruent numbers

Hamid Reza Abdolmalki and Farzali Izadi
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 1, Pages 14–24
DOI: 10.7546/nntdm.2019.25.1.14-24
Full paper (PDF, 191 Kb)

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Authors and affiliations

Hamid Reza Abdolmalki
Department of Mathematics, Faculty of Science
Azarbaijan Shahid Madani University
Tabriz 53751-71379, Iran

Farzali Izadi
Department of Mathematics, Faculty of Science
Azarbaijan Shahid Madani University
Tabriz 53751-71379, Iran

Abstract

In this note, we introduce elementary methods for obtaining new families of congruent numbers (CNs). By our methods, we can produce other CNs when one or two CNs are given. Also, we use some of the Pell equations (PEs) for getting some families of CNs. Up to now, it is not exactly determined which prime numbers of the form p = 8k + 1 are CNs. Among other things, we also introduce two simple methods to find some CNs of the forms p ≡ 1 (mod 8) and 2p where p is a prime number. By non-CNs and our methods, we also obtain some Diophantine equations (especially of degree 4), which have no positive solutions. In the end, we obtain a result on Heron triangles.

Keywords

• Congruent numbers
• Elliptic curves
• Rank
• Pythagorean triples
• Pell equations
• Heron triangles

2010 Mathematics Subject Classification

• 11G05
• 14H52
• 14G05
• 11E16
• 11D09

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