Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question

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.

share|improve this answer
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.