On the Diophantine equation \(x^4-q^4 = py^3\)
type: Item
instance of: scholarly article
title: On the Diophantine equation \(x^4-q^4 = py^3\)
zbMATH Open document ID: 1215.11027
DOI: 10.1216/RMJ-2010-40-3-995
author: Florian Luca
author: Alain Togbé
published in: Rocky Mountain Journal of Mathematics
publication date: 2010-07-23T00:00:00Z
review text: Let \(p\) and \(q\) be primes. The authors consider the Diophantine equation \[ x^4-q^4=py^3 \] and prove that for coprime integers \(x\) and \(y\) no solution exists. This is done by showing that a solution to this Diophantine equation corresponds to an integer point on certain elliptic curves. But no integer point of these elliptic curves yields a solution to the original Diophantine equation. Moreover, the authors also discuss the case of \(x\) and \(y\) have a common divisor. In this case the original Diophantine equation reduces to the equation \[ x^4-1=pq^2y^3. \] Under some technical restrictions on \(p\) and \(q\) the authors show that this Diophantine equation has no solution with \(y\not =0\) and conjecture that there exist only finitely many positive integer solutions \((x,y,p,q)\) with \(p,q\) prime. In this case the authors reduce the Diophantine equation to certain Thue equations.
reviewed by: Volker Ziegler
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 11D59
zbMATH DE Number: 5760104
zbMATH Keywords: Diophantine equations
zbMATH Keywords: elliptic curves
zbMATH Keywords: Thue equations
MaRDI profile type: Publication
full work available at URL: rmj-2010-40-3-995
OpenAlex ID: W2053663344
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description: scientific article; zbMATH DE number 5760104