Efficiently Certifying Non-Integer Powers

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 qp, such that a mod qp is a p-th non-residue. We use an effective version of the Chebotarev density theorem to estimate the density of such prime numbers qp. Keywords. Integer roots, integer powers, linear-time algorithm, bit complexity, Chebotarev density theorem. Subject classification. 11Y16 Number Theory: algorithms; complexity 68W30 Computer Science: symbolic computation and algebraic computation,

• Type: Journal
• Year: 2010
• Where: CC
• Authors: Erich Kaltofen, Mark Lavin
• Keywords: CC 2010, Chebotarev Density, Integer, Prime Numbers, System Software