# 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.

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let $a=2n\rightarrow a^2=4n^2\equiv 0 \bmod 4$ – Carry on Smiling Feb 21 '14 at 2:43
let $a=(2n+1)^2=4n^2+4n+1=4(n^2+n)+1\equiv1 \bmod 4$ – Carry on Smiling Feb 21 '14 at 2:44