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

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## Details

### 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 *y*^{2} = *x*^{3} – *A*^{2}*x* + *B*^{2}, 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 *y*^{2} = *x*(*x* – *a*^{2})(*x* + *c*^{2}) and *y*^{2} = *x*(*x* – *b*^{2})(*x* + *c*^{2}), 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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## Cite this paper

Baghalaghdam, Mehdi & Izadi, F. (2018). Some new families of positive-rank elliptic curves arising from Pythagorean triples. Notes on Number Theory and Discrete Mathematics, 24(3), 27-36, doi: 10.7546/nntdm.2018.24.3.27-36.