The least k-th power non-residue

Let p be a prime number and let k >= 2 be a divisor of p - 1. Norton proved that the least k-th power non-residue mod p is at most 3.9p(1/4) logp unless k = 2 and p equivalent to 3 (mod 4), in which case the bound is 4.7p(1/4) logp. By improving the upper bound in the Burgess inequality via a combinatorial idea, and by using some computing power, we improve the upper bounds to 0.9p(1/4) logp and 1.1p(1/4) logp, respectively. (C) 2014 Elsevier Inc. All rights reserved.