# An elementary proof of a result Ma and Chen

## TL;DR

This paper provides an elementary proof confirming Ma and Chen's result that a specific exponential Diophantine equation has only one positive integer solution under certain conditions, simplifying previous complex proofs.

## Contribution

The paper offers a new, elementary proof of Ma and Chen's theorem on the uniqueness of solutions to a particular exponential Diophantine equation.

## Key findings

- Confirmed the unique solution (2,2,2) under specified conditions
- Simplified the proof process compared to previous methods
- Validated the conjecture for cases where 4 does not divide the product mn

## Abstract

In 1956, Je$\acute{s}$manowicz conjectured that, for positive integers $m$ and $n$ with $m>n, \, \gcd(m,\, n)=1$ and $m\not\equiv n\pmod{2}$, the exponential Diophantine equation $(m^2-n^2)^x+(2mn)^y=(m^2+n^2)^z$ has only the positive integer solution $(x,\,y,\, z)=(2,\,2,\,2)$. Recently, Ma and Chen \cite{MC17} proved the conjecture if $4\not|mn$ and $y\ge2$. In this paper, we present an elementary proof of the result of Ma and Chen \cite{MC17}.