Pell's equations \(X^ 2-mY^ 2=-1,-4\) and continued fractions
type: Item
instance of: scholarly article
title: Pell's equations \(X^ 2-mY^ 2=-1,-4\) and continued fractions
zbMATH Open document ID: 0596.10013
DOI: 10.1016/0022-314X(86)90087-9
author: Pierre Kaplan
author: Kenneth S. Williams
published in: Journal of Number Theory
publication date: 1986-01-01T00:00:00Z
review text: Let \(m\equiv 1(mod 4)\) be a positive nonsquare integer such that the equation \(x^ 2-my^ 2=-1\) is solvable in integers x,y. The authors show that the equation \(x^ 2-my^ 2=-4\) is solvable in coprime integers x,y if and only if \(\ell (\sqrt{m})\equiv \ell ((1+\sqrt{m}))(mod 4),\) where \(\ell (\alpha)\) denotes the length of the period of the continued fraction expansion of the quadratic irrational \(\alpha\). Further congruences modulo 4 for \(\ell (\sqrt{m})\) and \(\ell ((1+\sqrt{m}))\) are given when \(m\equiv 5(mod 8)\).
Mathematics Subject Classification ID: 11D09
Mathematics Subject Classification ID: 11A55
zbMATH DE Number: 3959528
zbMATH Keywords: quadratic diophantine equation
zbMATH Keywords: Pell equation
zbMATH Keywords: period length
zbMATH Keywords: continued fraction expansion
zbMATH Keywords: congruences
reviewed by: G. Lomadze
MaRDI profile type: Publication
cites work: Q4154621
full work available at URL: 0022-314x(86)90087-9
OpenAlex ID: W1971846987
description: scientific article; zbMATH DE number 3959528