# Distribution of power residues and non-residues

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The distribution among the numbers of those values of for which the congruence  , is solvable (or unsolvable) in integers. Questions on the distribution of power residues and non-residues have been studied most fully in the case modulo a prime number . Let . Then the congruence is solvable for values of in the set and unsolvable for the remaining values (see Two-term congruence). However, comparatively little is known about how these values are distributed among the numbers .

The first results about the distribution of power residues were obtained by C.F. Gauss (see ) in 1796. From that time until the work of I.M. Vinogradov only isolated special results were obtained on questions concerning the distribution of power residues and non-residues. In 1915, Vinogradov (see ) proved a series of general results about the distribution of power residues and non-residues, and about primitive roots (cf. Primitive root) modulo among the numbers . In particular, he obtained the bound for the least quadratic non-residue , and the bound where is the number of distinct prime divisors of , for the least primitive root modulo .

In addition, he made a number of conjectures on the distribution of quadratic residues and non-residues (see Vinogradov hypotheses) which stimulated a series of investigations in this area. Thus, Yu.V. Linnik  proved that for sufficiently large , the number of prime numbers in the interval for which does not exceed a certain constant , depending only on . Thus, the prime numbers for which , if they exist at all, are met only very rarely. Another significant step in the work on Vinogradov's conjectures was the theorem of D.A. Burgess : For any fixed sufficiently small , the maximal distance between neighbouring quadratic non-residues satisfies the inequality In particular, one has In these inequalities, the constants , depend only on and not on . The proof of Burgess' theorem is very complicated; it is based on the Hasse–Weil theorem on the number of solutions of the hyper-elliptic congruence the proof of which requires techniques of abstract algebraic geometry. For a simple account of Burgess' theorem see , .

How to Cite This Entry:
Distribution of power residues and non-residues. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Distribution_of_power_residues_and_non-residues&oldid=14183
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article