The negative Pell equation and Pythagorean triples
type: Item
instance of: scholarly article
title: The negative Pell equation and Pythagorean triples
zbMATH Open document ID: 0971.11013
author: Aleksander Grytczuk
author: Florian Luca
author: M. Wójtowicz
published in: Proceedings of the Japan Academy. Series A
publication date: 2001-02-28T00:00:00Z
review text: Let \(d\) be a positive integer and \((A,B,C)\) be a primitive Pythagorean triple. In this paper, the authors prove that the Pell equation \(x^2-dy^2=-1\) is solvable in integers \(x\) and \(y\) if and only if there exist positive integers \(a,b\) and an \((A,B,C)\) such that \(d=a^2+b^2\) and \(|aA-bB|=1\). In this case, \(x=|aB+bA|\) and \(y=C\). Similarly, the authors also prove that the number \(\epsilon=(x+y\sqrt d)/2\), where \(x,y\) are relatively prime integers and \(x\) is odd, is the fundamental unit of \(\mathbb{Q}(\sqrt d)\) with the negative norm if and only if there exist positive integers \(a,b\) and an \((A,B,C)\) such that \(d=a^2+b^2\) and \(|aA-bB|=2\), and \((a,b)=(A,B)=1\). The method is elementary.
Mathematics Subject Classification ID: 11D09
Mathematics Subject Classification ID: 11R27
zbMATH DE Number: 1538532
zbMATH Keywords: Pell equation
zbMATH Keywords: Pythagorean triples
zbMATH Keywords: fundamental unit
reviewed by: Zhenfu Cao
MaRDI profile type: Publication
full work available at URL: pjaa.76.91
OpenAlex ID: W1986273466
Wikidata QID: Q114023237
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DOI: 10.3792/PJAA.76.91
description: scientific article; zbMATH DE number 1538532