For a real x > 1 and an integer g 6= 0,±1, an x-pseudopower to base g is a positive integer which is not a power of g yet is so modulo p for all primes p ≤ x. Let qg(x) denote the least such number. Improving a series of previous results we show that qg(x) ≤ exp(0.86092x) for sufficiently large x. The method is based on a combination of some bounds of exponential sums, estimates on the average behaviour of the multiplicative order of g modulo prime numbers and some results on the gaps between elements of multiplicative groups in residue rings. Mathematical Subject Classification: 11A07, 11L07,

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@inproceedings{BourgainOnTS, title={On the Smallest Pseudopower}, author={Jean Bourgain and Sergei Konyagin and Igor E. Shparlinski} }