We examine primitive roots modulo the Fermat number $F_m=2^{2^m}+1$. We show that an odd integer $n\ge 3$ 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 show that any factorization of any composite Fermat number $F_m={2^2}^m+1$ into two nontrivial factors can be expressed in the form $F_m=(k{2}^n+1)(\ell 2^n+1)$ for some odd $k$ and $\ell ,k\ge 3,\ell \ge 3$, and integer $n\ge m+2,3n<2^m$. We prove that the greatest common divisor of $k$ and $\ell$ is 1, $k+\ell \equiv 0mod{2}^n,max(k,\ell )\ge F_{m-2}$, and either $3|k$ or $3|\ell$, i.e., $3h{2}^{m+2}+1|F_m$ for an integer $h\ge 1$. Factorizations of $F_m$ into more than two factors are investigated as well. In particular, we prove that if $F_m={(k{2}^n+1)}^2(\ell 2^j+1)$ then $j=n+1,3|\ell$ and $5|\ell$.

The authors examine the frequency distribution of second-order recurrence sequences that are not $p$-regular, for an odd prime $p$, and apply their results to compute bounds for the frequencies of $p$-singular elements of $p$-regular second-order recurrences modulo powers of the prime $p$. The authors’ results have application to the $p$-stability of second-order recurrence sequences.

A positive integer $n$ is said to be a Jordan-Pólya number if it can be written as a product of factorials. We obtain non-trivial lower and upper bounds for the number of Jordan-Pólya numbers not exceeding a given number $x$.

A power digraph modulo $n$, denoted by $G(n,k)$, is a directed graph with $Z_n=\{0,1,\cdots ,n-1\}$ as the set of vertices and $E=\{(a,b):a^k\equiv b(modn)\}$ as the edge set, where $n$ and $k$ are any positive integers. In this paper we find necessary and sufficient conditions on $n$ and $k$ such that the digraph $G(n,k)$ has at least one isolated fixed point. We also establish necessary and sufficient conditions on $n$ and $k$ such that the digraph $G(n,k)$ contains exactly two components. The primality of Fermat number is also discussed.