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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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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.

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How do we prove perfect squares are 0 mod 4 or 1 mod 4 then? – user129818 Feb 20 '14 at 17:27
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

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