A generalization of Selfridge's question

Devendra Prasad

paper

A generalization of Selfridge's question

arXiv:1906.03433

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 $M_{n} (R)$ .