Geodesic
On the Distribution of Pseudopowers
Canadian journal of mathematics, Tome 62 (2010) no. 3, pp. 582-594

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DOI

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.
DOI : 10.4153/CJM-2010-020-4
Mots-clés : 11A07, 11L07, 11N36 
Konyagin, Sergei V.; Pomerance, Carl; Shparlinski, Igor E. On the Distribution of Pseudopowers. Canadian journal of mathematics, Tome 62 (2010) no. 3, pp. 582-594. doi: 10.4153/CJM-2010-020-4
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     title = {On the {Distribution} of {Pseudopowers}},
     journal = {Canadian journal of mathematics},
     pages = {582--594},
     year = {2010},
     volume = {62},
     number = {3},
     doi = {10.4153/CJM-2010-020-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2010-020-4/}
}
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