A generalization of Selfridge's question

Abstract

Selfridge asked to investigate the pairs $(m,n)$ of natural numbers for which $2^m - 2^n$ divides $x^m - x^n$ for all integers $x.$ This question was answered by different mathematicians by showing that there are only finitely many such pairs. Let $R$ be the ring of integers of a number field $\K$ and $M_n(R)$ be ring of all $n \times n$ matrices over $R$. In this article, we prove a generalization of Selfridge's question in the case of $M_n(R)$.