The least k-th power non-residue

Let p   be a prime number and let k≥2k≥2 be a divisor of p−1p−1. Norton proved that the least k-th power non-residue mod p   is at most 3.9p1/4log⁡p3.9p1/4log⁡p unless k=2k=2 and p≡3(mod4), in which case the bound is 4.7p1/4log⁡p4.7p1/4log⁡p. 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 0.9p1/4log⁡p0.9p1/4log⁡p and 1.1p1/4log⁡p1.1p1/4log⁡p, respectively.