The group of quasi-Pythagorean primitive triples
type: Item
instance of: scholarly article
title: The group of quasi-Pythagorean primitive triples
zbMATH Open document ID: 0938.11014
DOI: 10.1007/BF02857305
author: Natalia Baldisserri
publication date: 1999-10-25T00:00:00Z
review text: Let \(m>1\) be a square-free integer. The set of primitive solutions of the diophantine equation \(x^2+my^2=z^2\) has a natural structure of an abelian group induced by the multiplicative group \({\mathbb Q}(\sqrt{-m})^*\). Denote this group by \(G_m\), and call it the group of quasi-Pythagorean primitive triples. Generalizing a result proposed in \textit{E. J. Eckert} [Math. Mag. 57, 22-27 (1984; Zbl 0534.10010)], the author shows that \(G_m\) is torsion-free, except for \(m=3\) (the torsion corresponds to the roots of unity in \({\mathbb Q}(\sqrt{-m})^*\) modulo \(\{\pm 1\}\)). For a finite number of cases, namely when each genus of the quadratic forms with discriminant \(-4m\) consists of a single class, the author also shows that \(G_m\) is a free abelian group of infinite rank.
reviewed by: Roberto Dvornicich
Mathematics Subject Classification ID: 11D09
Mathematics Subject Classification ID: 20K99
zbMATH DE Number: 1339487
zbMATH Keywords: quadratic diophantine equations
zbMATH Keywords: torsion-free abelian group
zbMATH Keywords: free abelian group of infinite rank
MaRDI profile type: Publication
cites work: Q4693156
cites work: The Group of Primitive Pythagorean Triangles
cites work: The group of Pythagorean triples in number fields
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 1339487