  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 $u < 100$.
