An elementary proof of a result Ma and Chen

Abstract

In 1956, Je$\acute{s}$manowicz conjectured that, for positive integers $m$ and $n$ with $m>n,\gcd (m,n)=1$ and $m\not\equiv n(\mathrm{mod}2)$, 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 $4\nmid mn$ and $y\ge 2$. In this paper, we present an elementary proof of the result of Ma and Chen \cite{MC17}.