Generalizing the Lehmer's totient problem

TL;DR

This paper explores a group-theoretical analogue of Lehmer's totient problem, proposing a generalization and discussing properties of automorphism counts in finite groups, extending the scope of the original number theory question.

Contribution

It introduces a new group-theoretical problem analogous to Lehmer's totient problem and proposes a broader generalization of the original conjecture.

Findings

Identifies a group-theoretical analogue involving automorphism counts

Proposes a new generalization of Lehmer's totient problem

Discusses properties of potential solutions in the group context

Abstract

An important unsolved question in number theory is the Lehmer's totient problem that asks whether there exists any composite number $n$ such that $\varphi(n)\mid n-1$, where $\varphi$ is the Euler's totient function. It is known that if any such $n$ exists, it must be odd, square-free, greater that $10^{30}$, and divisible by at least $15$ distinct primes. Such a number must be also a Carmichael number. In this short note, we discuss a group-theoretical analogous problem involving the function that counts the number of automorphisms of a finite group. Another way to generalize the Lehmer's totient problem is also proposed.