On runs of residues

Abstract

According to a theorem of Alfred Brauer [1] all sufficiently large primes have runs of I consecutive integers that are kth power residues, wher-e k and I are arbitrarily given integers. In this paper we consider the question of the first appearance of such runs. Let p be a sufficiently large prime and let r = r(k, 1, p) be the least positive integer such that (1) r, r+ 1, r+ 2,*, r+l-1 are all congruent modulo p to kth powers of integers > 0. It is natural to ask, when k and I are given, how large is this minimum r and are there primes p for which r is arbitrarily large? If we let A(k, 1) = lim sup r(k, 1, p) p 00 c