On The Exceptional solutions of Je\'smanowicz' conjecture

TL;DR

This paper proves that for large enough parameters, the Diophantine equation related to Je'smanowicz' conjecture has no exceptional solutions when all exponents are even, advancing understanding of the conjecture.

Contribution

It establishes explicit bounds for parameters ensuring the absence of certain solutions, improving previous results and supporting Je'smanowicz' conjecture.

Findings

No exceptional solutions for large m when all exponents are even.

Explicit constant c(n) depending on n is provided.

Supports Je'smanowicz' conjecture under specific divisibility conditions.

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.