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Math Help - Sum of 3 squares

  1. #1
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    Sum of 3 squares

    I apologize for re-posting this question, but someone answered yesterday without reading the entire problem and nobody else seems to be looking at it. Thanks.

    Prove that if a prime number is a sum of three squares of different primes, then one of the primes must be equal to 3.
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Here's a thought:

    Try showing  p^2+q^2+r^2 is composite when  p,q,r>3 are prime.
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  3. #3
    MHF Contributor chiph588@'s Avatar
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    So we have  p=a^2+b^2+c^2 where  p,a,b,c are all distinct primes.

    Assume  a,b,c\neq3 \implies (a,3)=(b,3)=(c,3)=1\implies a^2\equiv b^2\equiv c^2\equiv1\bmod{3}

    This means  p\equiv 1+1+1\equiv0\bmod{3}\implies 3\mid p , which is a contradiction.
    Last edited by chiph588@; April 24th 2010 at 01:36 PM.
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