If $a$ and $b$ are odd then $a^2+b^2$ is not a perfect square 
Prove if $a$ and $b$ are odd then $a^2+b^2$ is not a perfect square. 

We have been learning proof by contradiction and were told to use the Euclidean Algorithm.
I have tried it both as written and by contradiction and can't seem to get anywhere.

Let $a=2m+1$ and $b=2n+1$. 
Assume $a^2+b^2=k^2$. Then:
$$(2m+1)^2+(2n+1)^2=k^2 \iff \\
4(m^2+m+n^2+n)+2=k^2 \iff \\
4(m^2+m+n^2+n)+2=(2r)^2 \iff \\
2(m^2+m+n^2+n)+1=2r^2 \iff \\
2s+1=2r^2,$$
which is a contradiction. Hence, the assumption $a^2+b^2=k^2$ is false. 
