ON THE CONJECTURE OF JEŚMANOWICZ CONCERNING PYTHAGOREAN TRIPLES

• Published in: Bulletin of the Australian Mathematical Society Vol. 80; no. 3; pp. 413 - 422
• Main Author: MIYAZAKI, TAKAFUMI
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.12.2009
• Subjects: exponential Diophantine equations; generalized Fermat equations; Integers; primary 11D61; Pythagorean triples; secondary 11D41; exponential Diophantine equations; Pythagorean triples; generalized Fermat equations; primary 11D61; secondary 11D41
• ISSN: 0004-9727; 1755-1633
• DOI: 10.1017/S0004972709000471

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.