On the diophantine equation \(x^p-x=y^q-y\)
type: Item
instance of: scholarly article
title: On the diophantine equation \(x^p-x=y^q-y\)
zbMATH Open document ID: 0949.11022
DOI: 10.5565/PUBLMAT_43199_08
published in: Publicacions Matemàtiques
publication date: 1999-10-25T00:00:00Z
full work available at URL: https://eudml.org/doc/41357
review text: The authors consider the diophantine equation \(x^p-x= y^q-y\) in integers \((x,p,y,q)\). They show that this equation has only finitely many solutions when some values among \(x\), \(y\), \(p\) and \(q\) are fixed. For example using the result of Davenport, Lewis, and Schinzel they show that for a given \(p\) and \(q\) with \(2\leq p< q\), the above equation has only finitely many integral solutions. Assuming the \(abc\)-conjecture they show that \(p\) and \(q\) are bounded. Thus under the \(abc\)-conjecture, the above equation has only finitely many solutions. Further they show that the curve \(C\) defined by the equation has only finitely many rational points \((x,y)\) when \(2\leq p< q\) and \(q\geq 4\). In the special case \(p=2\) and \(y\) a prime power they solved the equation completely.
Mathematics Subject Classification ID: 11D61
zbMATH DE Number: 1331812
zbMATH Keywords: exponential diophantine equations
zbMATH Keywords: \(S\)-integers
zbMATH Keywords: integral solutions
zbMATH Keywords: rational solutions
zbMATH Keywords: finiteness theorem
zbMATH Keywords: \(abc\)-conjecture
zbMATH Keywords: rational points
author: Maurice Mignotte
author: Attila Pethő
reviewed by: N. Saradha
MaRDI profile type: Publication
OpenAlex ID: W2262385164
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description: scientific article; zbMATH DE number 1331812