We estimate exponential sums with the Fermat-like quotients $$ f_g(n) = \frac{ g^{ n-1} - 1}{ n} \mand h_g(n)=\frac{ g^{ n-1}-1}{ P(n)}, $$ 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 ${ \widetilde h}_g(n) = gh_g(n)$.
Categories: math.NT