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Math Help - Quadratic Residues

  1. #1
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    Quadratic Residues

    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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  2. #2
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    Re: Quadratic Residues

    Answer :- f we consider the sum:

    ∑ i^2, where 1 ≤ i ≤ (p-1)/2 (note: only half of the squared numbers)

    It is possible to show (by induction) that this sum can be calculated as

    ∑ i^2 = (p-1)*p*(p+1)/24

    Hence the sum of the quadratic residues can be divided by p. (Why does it suffice to sum the first half of the squared numbers between 1 and p-1?).

    On the other hand we have the sum:

    ∑ k, where 1 ≤ k ≤ p-1 (all residues mod p both quadratic and non-)

    This is known as

    ∑ k = (p-1)*p/2

    And is clearly divisible by p. Since the sum of the quadratic residues is part of this sum and is divisible by p then the rest of this sum, namely the quadratic non residues, has to be divisible by p.

    and also know about How to Factor a Number.
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  3. #3
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    Re: Quadratic Residues

    Hi,
    brosnan123 has given a complete proof, almost. At the end he sums all the elements instead of the squares of all the elements. Here is his proof fleshed out a bit:

    Quadratic Residues-mhfnumbertheory1.png
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