Uniform Distribution of Fractional Parts Related to Pseudoprimes

Abstract

Abstract.We estimate exponential sums with the Fermat-like quotientswhere g and n are positive integers, n is composite, and P (n) is the largest prime factor of n. Clearly, both fg (n) and hg (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 n. Nevertheless, our bounds imply that the fractional parts ﹛ fg (n)﹜ and ﹛hg (n)﹜ are uniformly distributed, on average over g for fg (n), and individually for hg (n). We also obtain similar results with the functions and .