On Pythagorean elliptic curves
type: Item
instance of: scholarly article
title: On Pythagorean elliptic curves
zbMATH Open document ID: 0817.11030
author: Norio Adachi
published in: Proceedings of the Japan Academy. Series A
publication date: 1995-02-16T00:00:00Z
review text: For a primitive Pythagorean triple \((a,b,c)\) with \(a\) even, let \(E= E(a,b,c)\) be the elliptic curve \(y^ 2= x(x- a^ 2)(x- c^ 2)\). The author gives necessary and sufficient conditions for \(E/\mathbb{Q}\) to have non-zero rank. Said conditions are expressed in terms of the existence of non-zero solutions to a system of two diophantine equations of degree 2 and 4. This equivalence is used to show that there exist infinitely many curves \(E/\mathbb{Q}\) of rank \(\geq 1\).
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11D25
zbMATH DE Number: 723975
zbMATH Keywords: primitive Pythagorean triple
zbMATH Keywords: elliptic curve
zbMATH Keywords: non-zero rank
zbMATH Keywords: system of two diophantine equations of degree 2 and 4
reviewed by: Roelof J. Stroeker
MaRDI profile type: Publication
full work available at URL: pjaa.70.226
OpenAlex ID: W2004820874
Wikidata QID: Q114023242
cites work: Q3712374
cites work: Q3743417
cites work: Q3346401
cites work: Q5582375
cites work: Q3710637
cites work: Q4004187
DOI: 10.3792/PJAA.70.226
description: scientific article; zbMATH DE number 723975