The Distribution of Quadratic Residues and Non-residues

1. If p is a prime other than 2, half of the numbers 1, 2, ..., p-l are quadratic residues (modp) 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 ZJJ)\pi+K (3) This implies, in particular, that the maximum number of consecutive quadratic residues or non-residues (modp) is O(p i+i)for large p. Previously it was knownf only that the number is O(p 112). Theorem 1 enables me to improve on Vinogradov's estimate § for the magnitude of the least (positive) quadratic non-residue (modp). Using * G. Polya, " t)ber die Verteilung der quadratischen Eeste und Nichtreste ", Gottinger NachrichUn (1918), 21-29. \ I. M. Vinogradov, " Sur la distribution des rfeidus et des non-r6sidus des puissances ",