A Note on Je{\'s}manowicz' Conjecture for Non-primitive Pythagorean Triples

TL;DR

This paper investigates Je{\'s}manowicz' conjecture on Pythagorean triples, providing a unified proof of existing results and establishing the conjecture's validity for all prime u less than 100.

Contribution

It offers a simplified proof of Le-Yang-Fu's theorem and verifies Je{\'s}manowicz' conjecture for specific prime cases, advancing understanding of the conjecture.

Findings

Unified proof of Le-Yang-Fu theorem.

Necessary conditions for exceptional solutions when v=2.

Verification of the conjecture for prime u<100.

Abstract

Let $(a, b, c)$ be a primitive Pythagorean triple parameterized as $a=u^2-v^2,\ b=2uv,\ c=u^2+v^2$,\ where $u>v>0$ are co-prime and not of the same parity. In 1956, L. Je{\'s}manowicz conjectured that for any positive integer $n$, the Diophantine equation $(an)^x+(bn)^y=(cn)^z$ has only the positive integer solution $(x,y,z)=(2,2,2)$. In this connection we call a positive integer solution $(x,y,z)\ne (2,2,2)$ with $n>1$ exceptional. In 1999 M.-H. Le gave necessary conditions for the existence of exceptional solutions which were refined recently by H. Yang and R.-Q. Fu. In this paper we give a unified simple proof of the theorem of Le-Yang-Fu. Next we give necessary conditions for the existence of exceptional solutions in the case $v=2,\ u$ is an odd prime. As an application we show the truth of the Je{\'s}manowicz conjecture for all prime values $u < 100$.