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Math Help - Sum of Two Squares theorem

  1. #1
    Senior Member Pinkk's Avatar
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    Sum of Two Squares theorem

    If m is odd and if every prime dividing m is congruent to 1 modulo 4, prove that m can be written as the sum of two squares m = a^{2} + b^{2} where gcd(a,b) = 1.

    So it's easy enough to prove that it is the sum of two squares (the product of two primes that be written as the sum of two squares is also the sum of two squares), but I don't know how to prove that the gcd would be 1. A follow up question is , if m is even and m/2 and every prime dividing m/2 is congruent to 1 modulo 4, prove that m can be wrtten as a sum of two squares m= a^{2} + b^{2} with gcd(a,b)=1. Thanks
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  2. #2
    Senior Member Pinkk's Avatar
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    Re: Sum of Two Squares theorem

    So I've tried using induction on p but I still can't get the fact that the two squares are relatively prime.
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  3. #3
    Super Member TheChaz's Avatar
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    Re: Sum of Two Squares theorem

    Firstly, how exactly did you show that m = a^2 + b^2?
    Secondly, are you allowed to use the Gaussian integers?
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  4. #4
    Senior Member Pinkk's Avatar
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    Re: Sum of Two Squares theorem

    Well, we showed m = a^{2}+b^{2} by using the fact that any prime congruent 1 modulo 4 can be written as a sum of two squares, and a product of sums of squares is a sum of squares as well. And no, we haven't covered those in class.
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