If p is a prime other than 2, half of the numbers
1, 2, … p—1
are quadratic residues (mod p) and the other half are quadratic non-residues. Various questions have been proposed concerning the distribution of the quadratic residues and non-residues for large p, but as yet only very incomplete answers to these questions are known. Many of the known results are deductions from the inequality
found independently by Pólya and Vinogradov, the symbol being Legendre's symbol of quadratic character.
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