Connectedness of digraphs from quadratic polynomials

Abstract

Suppose that $f\left(x\right)=x\left(x-k\right)$, where $k$ is an odd positive integer. First, an infinite digraph $G_k=\left(V,E\right)$ is defined, where the vertex set is $V=\mathbb{Z}$ and the edge set is $E=\left\{\left(x,y\right)\mid x,y\in \mathbb{Z},f\left(x\right)=f\left(2y\right)\right\}$. Then the following results are proved: if $k=1$, then the digraph $G_k$ is weakly connected; if $p$ is a safe prime, i.e., both $p$ and $q=\left(p-1\right)∕2$ are primes, then the number $w_p$ of weakly connected components of the digraph $G_p$ is 2. Finally, a conjecture that there are infinitely many primes $p$ such that $w_p=2$ is presented.