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.
All squares are congruent to 1 or 0 mod 4. If they are odd they are congruent to 1 mod 4. therefore the sum of two odd squares is congruent to 2 mod 4. Thus not a square.