Efficiently Certifying Non-Integer Powers
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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 pth non-residue. We use an effective version of the Chebotarev density theorem to estimate the density of such prime numbers q p .
Keywords.Integer roots integer powers linear-time algorithm bit complexity Chebotarev density theorem
Subject classification.11Y16 68W30
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