Elementary results for certain Diophantine equations
type: Item
instance of: scholarly article
title: Elementary results for certain Diophantine equations
zbMATH Open document ID: 1067.11014
author: Pierre Samuel
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2005-06-30T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/226472
review text: In a series of papers (1942--1954) W. Ljunggren proved that for a given positive integer \(a\), the Diophantine equations \(x^4-ay^2=1\) and \(x^2-ay^4=1\) have at most two solutions in pairs of positive integers \((x,y)\). The proofs of Ljunggren used complicated studies of units of certain quartic number fields. The author considers the problem: determine the number of such solutions. For example, using only elementary methods he solves completely this problem for the family of equations \(x^4-ay^2=1\) when \(a\) is an odd prime number or twice an odd prime number. The paper also contains many other results on similar equations and also on \(x^2-ay^4=1\).
reviewed by: Maurice Mignotte
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 11D45
zbMATH DE Number: 2184604
zbMATH Keywords: quartic Diophantine equations
MaRDI profile type: Publication
OpenAlex ID: W2326588895
cites work: THE DIOPHANTINE EQUATION x4−Dy2=1
cites work: The Diophantine equation x⁴ - Dy² = 1, II
cites work: Q5828064
cites work: Q4321745
DOI: 10.5802/JTNB.379
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description: scientific article; zbMATH DE number 2184604