  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 $p$.

Number Theory