Pairs of Consecutive Power Residues

Abstract

Until recently none of the numerous papers on the distribution of quadratic and higher power residues was concerned with questions of the following sort: Let k and m be positive integers. According to a theorem of Brauer (1), for every sufficiently large prime p there exist m consecutive positive integers r, r + 1 , . . . , r + m — 1, each of which is a kth power residue of p. Let r(k, m, p) denote the least such r.