Distribution of power residues and non-residues
The distribution among the numbers of those values of
for which the congruence
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, 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 [1]) 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 [2]) 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
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for the least quadratic non-residue , and the bound
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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 [3] 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 [4]: For any fixed sufficiently small
, the maximal distance
between neighbouring quadratic non-residues satisfies the inequality
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In particular, one has
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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
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the proof of which requires techniques of abstract algebraic geometry. For a simple account of Burgess' theorem see [5], [6].
References
[1] | C.F. Gauss, "Untersuchungen über höhere Arithmetik" , A. Maser (1889) (Translated from Latin) |
[2] | I.M. Vinogradov, "Selected works" , Springer (1985) (Translated from Russian) |
[3] | Yu.V. Linnik, Dokl. Akad. Nauk SSSR , 36 (1942) pp. 131 |
[4] | D.A. Burgess, "The distribution of quadratic residues and non-residues" Mathematika , 4 : 8 (1957) pp. 106–112 |
[5] | S.A. Stepanov, "Constructive methods in the theory of equations over finite fields" Proc. Steklov Inst. Math. , 132 (1975) pp. 271–281 Trudy Mat. Inst. Steklov. , 132 (1973) pp. 237–246 |
[6] | A.A. Karatsuba, "Character sums and primitive roots in finite fields" Soviet Math.-Dokl. , 9 : 3 (1968) pp. 755–757 Dokl. Akad. Nauk SSSR , 180 : 6 (1968) pp. 1287–1289 |
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=23814