## Abstract

We estimate exponential sums with the Fermat-like quotients f _{g}(n)=g^{n-1}-1/n and h_{g}=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 &flm;_{g}(n) = gf_{g}(n) and h_{g}(n) - gh _{g}(n).

Original language | English |
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Pages (from-to) | 481-502 |

Number of pages | 22 |

Journal | Canadian Journal of Mathematics |

Volume | 61 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jun 2009 |