{
  "id": "1104.3909",
  "version": "v1",
  "published": "2011-04-20T00:31:23.000Z",
  "updated": "2011-04-20T00:31:23.000Z",
  "title": "Fermat quotients: Exponential sums, value set and primitive roots",
  "authors": [
    "Igor E. Shparlinski"
  ],
  "doi": "10.1112/blms/bdr058",
  "categories": [
    "math.NT"
  ],
  "abstract": "For a prime $p$ and an integer $u$ with $\\gcd(u,p)=1$, we define Fermat quotients by the conditions $$ q_p(u) \\equiv \\frac{u^{p-1} -1}{p} \\pmod p, \\qquad 0 \\le q_p(u) \\le p-1. $$ D. R. Heath-Brown has given a bound of exponential sums with $N$ consecutive Fermat quotients that is nontrivial for $N\\ge p^{1/2+\\epsilon}$ for any fixed $\\epsilon>0$. We use a recent idea of M. Z. Garaev together with a form of the large sieve inequality due to S. Baier and L. Zhao, to show that on average over $p$ one can obtain a nontrivial estimate for much shorter sums starting with $N\\ge p^{\\epsilon}$. We also obtain lower bounds on the image size of the first $N$ consecutive Fermat quotients and use it to prove that there is a positive integer $n\\le p^{3/4 + o(1)}$ such that $q_p(n)$ is a primitive root modulo $p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-04-20T00:31:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "11L40",
      "11N35"
    ],
    "keywords": [
      "exponential sums",
      "value set",
      "consecutive fermat quotients",
      "large sieve inequality",
      "define fermat quotients"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1104.3909S"
    }
  }
}