Residues and Non-ResiduesDate: 05/04/2003 at 15:27:05 From: Matt Subject: Number theory If p > 3, show that p divides the sum of its quadratic residue. I was able to show it with small primes like 13 and 17, but couldn't do it with general p. Any suggestions? Date: 05/05/2003 at 06:14:50 From: Doctor Jacques Subject: Re: Number theory Hi Matt, Let R be the sum of the (non-zero) quadratic residues, and N the sum of the non-residues. We have: N + R = 0 (mod p) (Do you see why?) For an odd prime p, there are (p-1)/2 residues and (p-1)/2 non- residues. If p > 3, p >= 5, and there are at least two non-residues. We can thus always pick a non-residue n <> p-1. If we multiply a non-residue by a non-zero residue, the product is a non-residue. This means that the set: {nr | r <> 0 is a residue} is exactly the set of the non-residues, since the two sets have the same number of elements. We can therefore write: nR = N with n <> -1 (mod p) Can you continue from here? Please feel free to write again if you require further assistance. - Doctor Jacques, The Math Forum http://mathforum.org/dr.math/ |
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