Pseudorandomness and Dynamics of Fermat Quotients
Alina Ostafe, Igor E. Shparlinski · SIAM Journal on Discrete Mathematics · 2011
We obtain some theoretical and experimental results concerning various properties (the number of fixed points, image distribution, cycle lengths) of the dynamical system naturally associated with Fermat quotients acting on the set $\{0,\dots,p-1\}$. In particular, we improve the lower bound of Vandiver [Bull. Amer. Math. Soc., 22 (1915), pp. 61–67] on the image size of Fermat quotients on the above set (from $p^{1/2}-1$ to $(1+o(1))p(\log p)^{-2}$). We also consider pseudorandom properties of Fermat quotients such as uniform distribution and linear complexity.