Jeśmanowicz' conjecture on exponential Diophantine equations
type: Item
instance of: scholarly article
title: Jeśmanowicz' conjecture on exponential Diophantine equations
zbMATH Open document ID: 1266.11064
author: Takafumi Miyazaki
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2011-12-19T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.facm/1323705814
review text: Jeśmanowicz' conjecture is the following statement: If \(a\), \(b\), \(c\) are coprime positive integers such that \(a^2+b^2=c^2\) with even \(b\), then the exponential equation \(a^x+b^y=c^z\) has the only solution \((x,y,z)=(2,2,2)\) in positive integers. This paper contains various new results on this conjecture. In particular, the author proves that if the previous equation has a solution \((x,y,z)\) with \(x\) and \(z\) even then \(x/2\) and \(z/2\) are odd.
reviewed by: Maurice Mignotte
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11D41
Mathematics Subject Classification ID: 11J86
zbMATH DE Number: 5989420
zbMATH Keywords: exponential Diophantine equations
zbMATH Keywords: generalized Fermat equations
zbMATH Keywords: Pythagorean triples
zbMATH Keywords: Baker's theory
MaRDI profile type: Publication
OpenAlex ID: W1985405758
Wikidata QID: Q122979622
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DOI: 10.7169/FACM/1323705814
description: scientific article; zbMATH DE number 5989420