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arXiv 1104.3909math.NT
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费马定理：指数和、值集与原根

Fermat quotients: Exponential sums, value set and primitive roots

Igor E. Shparlinski

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摘要：

对于素数$p$和具有$\gcd(u,p)=1$的整数$u$，我们通过条件$$ q_p(u) \equiv \frac{u^{p-1} -1}{p} \pmod p, \qquad 0 \le q_p(u) \le p-1. $$ D定义费马行列式。R. Heath-Brown给出了$N$连续费马行列式的指数和的一个界，该界对于任何固定的$ε>0$，对$N\ge p^{1/2+ε}$是非平凡的.我们使用M的一个最近的想法。Z. Garaev和S. Baier和L. Zhao，证明了在$p$上平均可以得到从$N\ge p^ε$开始的更短和的非平凡估计。我们还得到了第一个$N$连续费马行列式的图像大小的下界，并利用它证明了存在正整数$n\le p^{3/4 + o(1)}$使得$q_p(n)$是模$p$的原根.

英文摘要：

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 $N\ge p^{1/2+ε}$ for any fixed $ε>0$. 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 $N\ge p^ε$. 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 $p$.

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