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Math Help - Primes of the form 4n+3

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    Primes of the form 4n+3

    Let p be a prime of the form 4n+3. Prove that there exists integers a,b such that 0 < a,b \leq p such that p \mid (a^{2}+b^{2})
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    Quote Originally Posted by Chandru1 View Post
    Let p be a prime of the form 4n+3. Prove that there exists integers a,b such that 0 < a,b \leq p such that p \mid (a^{2}+b^{2})

    As you put it the problem is trivial: choose a,b=p\Longrightarrow p\mid a^2+b^2=2p^2 , and if you meant 0<a,b<p then the problem is false, as p=3 shows.

    Tonio
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    MHF Contributor Bruno J.'s Avatar
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    Yeah, something makes me think that there are two mistakes in the statement of this problem! Because it is similar to the very interesting problem

    Let p be a prime of the form 4n+1. Show that there exist integers 0<a,b<p such that p|a^2+b^2.
    which is not a trivial problem at all.
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