Divisibility of Fermat Quotients

Tuesday, December 8, 2015 - 2:00pm
MSB 312
Igor Shparlinski (University of New South Wales)

We show that for a prime $p$ the smallest $a$ with $a^{p-1} \not \equiv 1 \pmod {p^2}$ 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 as $(\log p)^{5/3 + o(1)}$. 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.