  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}.
