Efficiently Certifying Non-Integer Powers

Abstract.

We describe a randomized algorithm that, given an integer a, produces a certificate that the integer is not a pure power of an integer in expected (log a)^1+o(1) bit operations under the assumption of the Generalized Riemann Hypothesis. The certificate can then be verified in deterministic (log a)^1+o(1) time. The certificate constitutes for each possible prime exponent p a prime number q _p , such that a mod q _p is a p^th non-residue. We use an effective version of the Chebotarev density theorem to estimate the density of such prime numbers q _p .