Fermat quotients: Exponential sums, value set and primitive roots

Igor E. Shparlinski*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

22 Citations (Scopus)


For a prime p and an integer u with gcd(u, p)=1, we define Fermat quotients by the conditions Heath-Brown has given a bound of exponential sums with N consecutive Fermat quotients that is non-trivial for N≥p 1/2+ε for any fixed ε>0. We use a recent idea of Garaev together with a form of the large sieve inequality due to Baier and Zhao to show that, on average over p, one can obtain a non-trivial estimate for much shorter sums starting with N≥p ε. We also obtain lower bounds on the image size of the first N consecutive Fermat quotients and use them to prove that there is a positive integer n≤p 3/4+o(1) such that qp(n) is a primitive root modulo p.

Original languageEnglish
Pages (from-to)1228-1238
Number of pages11
JournalBulletin of the London Mathematical Society
Issue number6
Publication statusPublished - Dec 2011

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