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

Posted in
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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