On the exponential Diophantine equation \(a^x+b^y=c^z\)
type: Item
instance of: scholarly article
title: On the exponential Diophantine equation \(a^x+b^y=c^z\)
zbMATH Open document ID: 1009.11026
author: Nobuhiro Terai
published in: Proceedings of the Japan Academy. Series A
publication date: 2003-05-01T00:00:00Z
review text: Let \(a\), \(b\), \(c\) be fixed positive integers such that \(\min(a,b,c)>1\) and \(\gcd(a,b,c)=1\). In this paper the author proves that if \(a,b,c\) satisfy some congruence conditions, then the equation \((*)\) \(a^x+b^y=c^z\) has only one positive integer solution \((x,y,z)\). For example, he proves that if \(a\equiv -1\pmod{b^2}\), \(b\) is an odd prime with \(b\equiv 3\pmod 4\) and \(a^2+b=c\), then \((*)\) has only the positive integer solution \((x,y,z)= (2,1,1)\).
Mathematics Subject Classification ID: 11D61
zbMATH DE Number: 1779509
zbMATH Keywords: congruence conditions
reviewed by: Le Maohua
MaRDI profile type: Publication
full work available at URL: pjaa.77.151
OpenAlex ID: W1972942094
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DOI: 10.3792/PJAA.77.151
description: scientific article; zbMATH DE number 1779509