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On the Distribution of Pseudopowers

Open Access article
 Printed: Jun 2010
  • Sergei V. Konyagin
  • Carl Pomerance
  • Igor E. Shparlinski
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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, 11L07, 11N36 show english descriptions Congruences; primitive roots; residue systems
Estimates on exponential sums
Applications of sieve methods
11A07 - Congruences; primitive roots; residue systems
11L07 - Estimates on exponential sums
11N36 - Applications of sieve methods

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