On the Lehmer's totient problem on Number Fields

TL;DR

This paper generalizes Lehmer's totient problem to algebraic number fields by introducing Lehmer numbers, which are algebraic integers satisfying the original problem's divisibility condition.

Contribution

It extends Lehmer's totient problem from integers to algebraic number fields and defines Lehmer numbers within this broader context.

Findings

Introduction of Lehmer numbers in algebraic number fields

Generalization of Lehmer's totient problem

Foundational framework for further research

Abstract

Lehmer's totient problem asks if there exists a composite number $d$ such that its totient divide $d-1$. In this article we generalize the Lehmer's totient problem in algebraic number fields. We introduce the notion of a Lehmer number. Lehmer numbers are defined to be the natural numbers which obey the Lehmer's problem in the ring of algebraic integers of a number field.