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# On the Distribution of Pseudopowers

Published:2009-12-04
Printed: Jun 2010
• Sergei V. Konyagin
• Carl Pomerance
• Igor E. Shparlinski
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## Abstract

An x-pseudopower to base g is a positive integer that is not a power of g, yet is so modulo p for all primes $ple x$. We improve an upper bound for the least such number, due to E.~Bach, R.~Lukes, J.~Shallit, and H.~C.~Williams. The method is based on a combination of some bounds of exponential sums with new results about the average behaviour of the multiplicative order of g modulo prime numbers.
 MSC Classifications: 11A07 - Congruences; primitive roots; residue systems 11L07 - Estimates on exponential sums 11N36 - Applications of sieve methods