Some congruence properties of the Pell equation
type: Item
instance of: scholarly article
zbMATH Open document ID: 1264.11019
author: Nathan Priddis
author: Jasbir S. Chahal
publication date: 2012-05-30T00:00:00Z
full work available at URL: 35-2.html
Mathematics Subject Classification ID: 11D09
zbMATH DE Number: 6039657
zbMATH Keywords: Pell equation
zbMATH Keywords: congruences
MaRDI profile type: Publication
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title: Some congruence properties of the Pell equation
published in: Annales des Sciences Mathématiques du Québec
review text: The group \(G(N)\) of the integer solutions \((x,y)\) of the Pell equation \(x^2-Ny^2=1\) can be viewed as a subgroup of \(\mathrm{GL}_2(\mathbb{Z})\). For an integer \(m>1\), we have the reduction group homomorphism \(\text{red}_m:G(N) \to \mathrm{GL}_2(\mathbb{Z}/m\mathbb{Z})\). In this paper, the authors study properties of the function \(g_N(m)=|\text{red}_m(G(N))|\). They show that if an odd prime \(p\) divides \(N\), then \(g_N(p) | 2p\), and if \(p\) does not divide \(N\), then \(g_N(p)\) divides \(p-1\) or \(p+1\).
reviewed by: Andrej Dujella
description: scientific article; zbMATH DE number 6039657