A note on a theorem of Ljunggren and the diophantine equations \(x^2-kxy^2+y^4=1,4\)
type: Item
instance of: scholarly article
title: A note on a theorem of Ljunggren and the diophantine equations \(x^2-kxy^2+y^4=1,4\)
zbMATH Open document ID: 0941.11012
published in: Archiv der Mathematik
publication date: 2000-08-10T00:00:00Z
review text: Let \(D\) denote a positive nonsquare integer. W. Ljunggren has shown that there are at most two solutions in positive integers \((x,y)\) to the diophantine equation \(x^2-Dy^2=1\), and that if two solutions \((x_1,y_1)\), \((x_2, y_2)\) exist, with \(x_1<x_2\), then \(x_1+y_1^2\sqrt D\) is the fundamental unit \(\varepsilon_D\) in the quadratic field \(\mathbb{Q}(\sqrt D)\), and \(x_2+y^2_2\sqrt D\) is either \(\varepsilon^2_D\) or \(\varepsilon^4_D\). The purpose of this note is twofold: Using a result of \textit{J. H. E. Cohn} [Acta Arith. 78, 401-403 (1997; Zbl 0870.11018)], Ljunggren's theorem is generalized. This generalization is then used to completely solve the diophantine equations in the title.
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 11J86
zbMATH DE Number: 1346254
zbMATH Keywords: quartic diophantine equations
zbMATH Keywords: quadratic diophantine equation
zbMATH Keywords: Ljunggren's theorem
author: P. G. Walsh
reviewed by: Edward L. Cohen
MaRDI profile type: Publication
full work available at URL: s000130050376
OpenAlex ID: W1998705282
DOI: 10.1007/S000130050376
description: scientific article; zbMATH DE number 1346254