The diophantine equation \(a^ x +b^ y= c^ z\). II
type: Item
instance of: scholarly article
title: The diophantine equation \(a^ x +b^ y= c^ z\). II
zbMATH Open document ID: 0842.11009
author: Nobuhiro Terai
published in: Proceedings of the Japan Academy. Series A
publication date: 1996-07-28T00:00:00Z
review text: [For part I see ibid. 70, 22-26 (1994; Zbl 0812.11024).] Let \(a= \pm m (m^4- 10m^2+ 5)\), \(b= 5m^4- 10m^2+ 1\) and \(c= m^2+ 1\), where \(m\) is an even rational integer. Assume that \(b\) is a prime and there exists an odd prime \(\ell\) such that \(ab\equiv 0\pmod \ell\) and 5 divides the order of \(c\pmod \ell\). In the paper under review it is proved that the diophantine equation \(a^x+ b^y= c^z\) has the only positive integral solution \((x, y, z) = (2, 2, 5)\). Some examples of \(m \leq 50\), \(a\), \(b\), \(c\), \(\ell\) satisfying the above conditions are given.
Mathematics Subject Classification ID: 11D61
zbMATH DE Number: 851721
zbMATH Keywords: exponential diophantine equation
zbMATH Keywords: positive integral solution
reviewed by: Dimitros Poulakis
MaRDI profile type: Publication
full work available at URL: pjaa.71.109
OpenAlex ID: W2372764239
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cites work: The diophantine equation \(a^ x+b^ y=c^ z\)
DOI: 10.3792/PJAA.71.109
description: scientific article; zbMATH DE number 851721