304 research outputs found
Pseudorandomness and Dynamics of Fermat Quotients
We obtain some theoretic 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 . We also consider pseudorandom properties of Fermat
quotients such as joint distribution and linear complexity
Fermat quotients: Exponential sums, value set and primitive roots
For a prime and an integer with , we define Fermat
quotients by the conditions D. R. Heath-Brown has given a bound of
exponential sums with consecutive Fermat quotients that is nontrivial for
for any fixed . 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 one can obtain a nontrivial
estimate for much shorter sums starting with . We also
obtain lower bounds on the image size of the first consecutive Fermat
quotients and use it to prove that there is a positive integer such that is a primitive root modulo
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