On Jeśmanowicz' conjecture concerning primitive Pythagorean triples. II
type: Item
instance of: scholarly article
title: On Jeśmanowicz' conjecture concerning primitive Pythagorean triples. II
zbMATH Open document ID: 1374.11056
published in: Acta Mathematica Hungarica
publication date: 2017-03-31T00:00:00Z
review text: The Jeśmanowicz conjecture states that for given \(m,n\in\mathbb{Z}\) with \(m>n>0,\gcd m,n=1\) and \(m\not\equiv n\pmod{2}\) the equation \[ (m^2-n^2)^x+(2mn)^y=(m^2+n^2)^z \] has the only solution \((x,y,z)=(2,2,2)\). In this paper the authors prove that the Jeśmanowicz conjecture is true under the following additional assumption: \(n\equiv 2\), \(m>72n\), at least one of the following three conditions is fulfilled -- \(n/2\) is a power of an odd prime; -- \(n/2\) has no prime factors congruent to 1 modulo 8; -- \(n/2\) is a square. As a corollary the authors show that this result implies that the Jeśmanowicz conjecture is true for every \(n<100\) for which \(n/2\) is odd. Part I, cf. the second author [J. Number Theory 141, 316-323 (2014; Zbl 1309.11030)].
reviewed by: Attila Bérczes
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11D41
zbMATH DE Number: 6699218
zbMATH Keywords: Pythagorean triple
zbMATH Keywords: exponential Diophantine equation
zbMATH Keywords: generalized Fermat equation
zbMATH Keywords: linear form in two logarithms
Wikidata QID: Q114227610
author: Takafumi Miyazaki
author: Nobuhiro Terai
MaRDI profile type: Publication
full work available at URL: s10474-015-0552-3
OpenAlex ID: W2187607260
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DOI: 10.1007/S10474-015-0552-3
description: scientific article; zbMATH DE number 6699218