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Uniform Distribution of Fractional Parts Related to Pseudoprimes

Published:2009-06-01
Printed: Jun 2009
• William D. Banks
• Moubariz Z. Garaev
• Florian Luca
• Igor E. Shparlinski
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Abstract

We estimate exponential sums with the Fermat-like quotients $$f_g(n) = \frac{g^{n-1} - 1}{n} \quad\text{and}\quad h_g(n)=\frac{g^{n-1}-1}{P(n)},$$ where $g$ and $n$ are positive integers, $n$ 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 $n$. 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 ${\widetilde f}_g(n) = gf_g(n)$ and ${\widetilde h}_g(n) = gh_g(n)$.
 MSC Classifications: 11L07 - Estimates on exponential sums 11N37 - Asymptotic results on arithmetic functions 11N60 - Distribution functions associated with additive and positive multiplicative functions