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Math Help - Sum of quadratic residues

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    Sum of quadratic residues

    a) Let p be a prime number with p>3. Prove that sum of quadratic residues modulo p is divisible by p.
    b) Let p be a prime number with p>5. Prove that the sum of the squares of the quadratic nonresidues modulo p is divisible by p.
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    Quote Originally Posted by mndi1105 View Post
    a) Let p be a prime number with p>3. Prove that sum of quadratic residues modulo p is divisible by p.
    b) Let p be a prime number with p>5. Prove that the sum of the squares of the quadratic nonresidues modulo p is divisible by p.
    Let a be a primitive root.
    Then all quadradic residues are given by 1,a^2,a^4,...,a^{p-2} i.e. even exponents.

    The sum is 1+a^2+a^4+...+a^{p-2} = \frac{1-a^p}{1-a^2} \equiv 0 (see how we use assumption p>3).

    In the second problem there is not need to have p>5. Let A be the residue sum and B be the non-residue sum. Then we know A + B = 0. Since p divides A it must means that p divides B.
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