## Abstract

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 language | English |
---|---|

Pages (from-to) | 1228-1238 |

Number of pages | 11 |

Journal | Bulletin of the London Mathematical Society |

Volume | 43 |

Issue number | 6 |

DOIs | |

Publication status | Published - Dec 2011 |