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On the Distribution of Pseudopowers

Published online by Cambridge University Press:  20 November 2018

Sergei V. Konyagin
Affiliation:
Department of Mechanics and Mathematics, Moscow State University, Moscow, 119992, Russia, e-mail: konyagin@ok.ru
Carl Pomerance
Affiliation:
Department of Mathematics, Dartmouth College, Hanover, NH 03755-3551, USA, e-mail: carlp@gauss.dartmouth.edu
Igor E. Shparlinski
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia, e-mail: igor@ics.mq.edu.au
Corresponding
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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 $p\,\le \,x$ . 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.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2010

References

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