304 research outputs found

    Pseudorandomness and Dynamics of Fermat Quotients

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    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 {0,...,pβˆ’1}\{0, ..., p-1\}. We also consider pseudorandom properties of Fermat quotients such as joint distribution and linear complexity

    Fermat quotients: Exponential sums, value set and primitive roots

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    For a prime pp and an integer uu with gcd⁑(u,p)=1\gcd(u,p)=1, we define Fermat quotients by the conditions qp(u)≑upβˆ’1βˆ’1p(modp),0≀qp(u)≀pβˆ’1. 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 NN consecutive Fermat quotients that is nontrivial for Nβ‰₯p1/2+Ο΅N\ge p^{1/2+\epsilon} for any fixed Ο΅>0\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 pp one can obtain a nontrivial estimate for much shorter sums starting with Nβ‰₯pΟ΅N\ge p^{\epsilon}. We also obtain lower bounds on the image size of the first NN consecutive Fermat quotients and use it to prove that there is a positive integer n≀p3/4+o(1)n\le p^{3/4 + o(1)} such that qp(n)q_p(n) is a primitive root modulo pp
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