Quadratic Residues

Definition (Quadratic Residue) Let   be a positive integer with     
    
    (i) If has a solution then is a quadratic residue of
    
    (ii) If does not have solution then is a quadratic nonresidue of
    

Example (Quadratic Residue) Determine the quadratic residues of
    Solution. We compute the squares of the positive integers less than 13 namely: and Therefore, the quadratic residues of 13 are and the quadratic nonresidues of 13 are

Proposition (Quadratic Residue) Let be an odd prime.

    (i)  If then either has no solutions or exactly two incongruent solutions modulo
    
    (ii) There are exactly quadratic residues of and quadratic nonresidues of among
    
    Proof. (i) If is a solution for then so is If then and so since is odd. Clearly, this can not happen and so if there are any solutions there are at least two incongruent solutions. If and are both solutions then and so either or Thus there can be only two incongruent solutions for if there are any at all.

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