Uniform Distribution of Fractional Parts Related to Pseudoprimes

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

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1 Citation (Scopus)


We estimate exponential sums with the Fermat-like quotients f g(n)=gn-1-1/n and hg=gn-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 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 h g(n). We also obtain similar results with the functions &flm;g(n) = gfg(n) and hg(n) - gh g(n).

Original languageEnglish
Pages (from-to)481-502
Number of pages22
JournalCanadian Journal of Mathematics
Issue number3
Publication statusPublished - Jun 2009


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