Type: Article

Uniform distribution of fractional parts related to pseudoprimes

William D. Banks; Moubariz Z. Garaev; Florian Luca; Igor E. Shparlinski;

Abstract:

We estimate exponential sums with the Fermat-like quotients f(g)(n) = g(n-1)-1/n and h(g)(n) = g(n-1)-1/P(n), where g and n are positive integers, it is composite, and P(n) is the largest prime factor of n. Clearly, both f(g)(n) and h(g)(n) are integers if n is a Fermat pseudoprime to base g, and if n is a Carmichael number, this is true for all g coprime to it. Nevertheless, our bounds imply that the fractional parts {f(g)(n) and {h(g)(n) are uniformly distributed, on average over g for f(g)(n), and individually for h(g)(n). We also obtain similar results with the functions (f) over tilde (g)(n) = gf(g)(n) and (h) over tilde (g)(n) = gh(g)(n).