Fermat–Euler Dynamical Systems and the Statistics of Arithmetics of Geometric Progressions

Let $n$ be an integer. A Fermat–Euler dynamical system acts on the set of mod-$n$ residues coprime to $n$ by multiplication by a constant (which is also coprime to $n$). We study the dependence of the period and the number of orbits of this dynamical system on $n$. Theorems generalizing Fermat's little theorem, as well as empirical conjectures, are given.