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Fermat Quotients (joint work with J. Bourgain, K. Ford and S. Konyagin)

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Speaker: 
Igor Shparlinski
Affiliation: 
Macquarie U, Sydney
Date: 
Fri, 2010-05-28 11:15 - 12:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

We show that for a prime $p$ the smallest $a$ with $a^{p-1}$ that is not congruent to 1 modulo ${p^2}$ does not exceed $(\log p)^{463/252 + o(1)}$ which improves the previous bound $O((\log p)^2)$ obtained by H. W. Lenstra in 1979. We also show that for almost all primes $p$ the bound can be improved to $(\log p)^{5/3 + o(1)}$. These results are based on a combination of various techniques including the distribution of smooth numbers, distribution of elements of multiplicative subgroups of residue rings, bound of Heilbronn exponential sums and a large sieve inequality with square moduli.

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