A group-theoretical approach to Lehmer's totient problem

TL;DR

This paper introduces a novel group-theoretical approach to Lehmer's totient problem, establishing new bounds and conditions that any potential counterexample must satisfy, significantly extending previous constraints.

Contribution

It develops a new method using recent group theory results to derive lower bounds on potential counterexamples to Lehmer's problem.

Findings

Any counterexample must satisfy n > 10^8171

The number of prime factors ω(n) must be at least 1991

If certain primes do not divide n, the factor k is at least 4

Abstract

Lehmer's totient problem asks whether there exists any composite number $n$ such that $\varphi(n) \, \mid \, (n-1)$, where $\varphi$ is Euler totient function. It is known that if any such $n$ exists, it must be Carmichael and $n > 10^{30}$. In this paper, we develop a new approach to the problem via some recent results in group theory related to a function $\psi$ (the sum of order of elements of a group) and show that if $k \varphi(n) = n-1$ for some integer $k$, then $k$ must be $\geq 3$, and actually, if $5, 7, 11, 13 \not | n$, $k \geq 4$. This implies that any counterexample must be such that $n > 10^{8171}$ and $\omega(n) \geq 1991$.