# A Note on Je{\'s}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>0are 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)^zhas 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>1exceptional. 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,\ uis an odd prime. As an application we show the truth of the Je{\'s}manowicz conjecture for all prime values u < 100