We examine primitive roots modulo the Fermat number Fm = 2 2m + 1. We show that an odd integer n ≥ 3 is a Fermat prime if and only if the set of primitive roots modulo n is equal to the set of quadratic non-residues modulo n. This result is extended to primitive roots modulo twice a Fermat number.
We assign to each pair of positive integers $n$ and $k\ge 2$ a digraph $G(n,k)$ whose set of vertices is $H=\{0,1,\dots ,n-1\}$ and for which there is a directed edge from $a\in H$ to $b\in H$ if $a^k\equiv b\pmod n$. We investigate the structure of $G(n,k)$. In particular, upper bounds are given for the longest cycle in $G(n,k)$. We find subdigraphs of $G(n,k)$, called fundamental constituents of $G(n,k)$, for which all trees attached to cycle vertices are isomorphic.