# Distribution of power residues and non-residues

2010 Mathematics Subject Classification: Primary: 11A15 Secondary: 11N69 [MSN][ZBL]

The distribution among the numbers $1,\ldots,m-1$ of those values of $x$ for which the congruence

$$y^n\equiv x\pmod{m},$$

$n>1$, 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 $p$. Let $q=\mathrm{gcd}(n,p-1)$. Then the congruence $y^n\equiv x\pmod{p}$ is solvable for $(p-1)/q$ values of $x$ in the set $1,\ldots,p-1$ and unsolvable for the remaining $(q-1)(p-1)/q$ values (see Two-term congruence). However, comparatively little is known about how these values are distributed among the numbers $1,\ldots,p-1$.

The first results about the distribution of power residues were obtained by C.F. Gauss (see [Ga]) 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 [Vi]) proved a series of general results about the distribution of power residues and non-residues, and about primitive roots (cf. Primitive root) modulo $p$ among the numbers $1,\ldots,p$. In particular, he obtained the bound

$$N_{\mathrm{min}} < p^{\frac{1}{2\sqrt{e}}}(\ln p)^2$$

for the least quadratic non-residue $N_{\mathrm{min}}$, and the bound

$$N^*_{\mathrm{min}}\leq 2^{2k}\sqrt{p}\ln p,$$

where $k$ is the number of distinct prime divisors of $p-1$, for the least primitive root $N_{\mathrm{min}}^*$ modulo $p$.

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 [Li] proved that for sufficiently large $N$, the number of prime numbers $p$ in the interval $[N^{\epsilon},N]$ for which $N_{\mathrm{min}}>p^{\epsilon}$ does not exceed a certain constant $C(\epsilon)$, depending only on $\epsilon>0$. Thus, the prime numbers $p$ for which $N_{\mathrm{min}}>p^{\epsilon}$, 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 [Bu]: For any fixed sufficiently small $\delta>0$, the maximal distance $d(p)$ between neighbouring quadratic non-residues satisfies the inequality

$$d(p)\leq A(\delta)p^{\frac{1}{4}+\delta}.$$

In particular, one has

$$N_{\mathrm{min}}\leq B(\delta)p^{\frac{1}{4\sqrt{e}}+\delta}.$$

In these inequalities, the constants $A(\delta)$, $B(\delta)$ depend only on $\delta$ and not on $p$. 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

$$y^2\equiv f(x)\pmod{p},$$

the proof of which requires techniques of abstract algebraic geometry. For a simple account of Burgess' theorem see [St], [Ka].

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=35720
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article