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

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{ś}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{ś}manowicz conjecture for all prime values $u<100$.

17 pages