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ON THE CONJECTURE OF JEŚMANOWICZ CONCERNING PYTHAGOREAN TRIPLES

Part of: Diophantine equations

Published online by Cambridge University Press:  02 July 2009

TAKAFUMI MIYAZAKI
TAKAFUMI MIYAZAKI*
Affiliation:
Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1, Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan (email: miyazaki-takafumi@ed.tmu.ac.jp)
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Abstract

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Let a,b,c be relatively prime positive integers such that a2+b2=c2 with b even. In 1956 Jeśmanowicz conjectured that the equation ax+by=cz has no solution other than (x,y,z)=(2,2,2) in positive integers. Most of the known results of this conjecture were proved under the assumption that 4 exactly divides b. The main results of this paper include the case where 8 divides b. One of our results treats the case where a has no prime factor congruent to 1 modulo 4, which can be regarded as a relevant analogue of results due to Deng and Cohen concerning the prime factors of b. Furthermore, we examine parities of the three variables x,y,z, and give new triples a,b,c such that the conjecture holds for the case where b is divisible by 8. In particular, to prove our results, we shall show an important result which asserts that if x,y,z are all even, then x/2,y/2,z/2 are all odd. Our methods are based on elementary congruence and several strong results on generalized Fermat equations given by Darmon and Merel.

Keywords

exponential Diophantine equationsPythagorean triplesgeneralized Fermat equations

MSC classification

Secondary: 11D61: Exponential equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 3 , December 2009 , pp. 413 - 422
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

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