On The Exceptional solutions of Je\'smanowicz' conjecture

Abstract

Let $(a,b,c)$ be a primitive Pythagorean triple. Set $a=m^2-n^2$,$b=2mn$, and $c=m^2+n^2$ with $m$ and $n$ positive coprime integers, $m>n $ and $ m \not \equiv n \pmod 2$. A famous conjecture of Je\'{s}manowicz asserts that the only positive solution to the Diophantine equation $a^x+b^y=c^z$ is $(x,y,z)(2,2,2).$ In this note, we will prove that for any $n>0$ there exists an explicit constant $c(n)>0$ such that if $m> c(n)$, then the above equation has no exceptional solution when all $x$,$y$ and $z$ are even. Our result improves that of Fu and Yang [11]. As an application, we will show that if $4 \mid\!\mid m$ and $m > c(n)$ Je\'{s}manowicz' conjecture holds.