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Prove if a and b are odd that 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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1 Answer 1

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 at 17:27
let $a=2n\rightarrow a^2=4n^2\equiv 0 \bmod 4$ –  Bananarama Feb 21 at 2:43
let $a=(2n+1)^2=4n^2+4n+1=4(n^2+n)+1\equiv1 \bmod 4$ –  Bananarama Feb 21 at 2:44

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