On the Lehmer Numbers, II

TL;DR

This paper investigates Lehmer numbers, establishing an Euler totient inequality, relating new parameters, and providing conditions under which certain numbers can or cannot be Lehmer numbers, advancing understanding of Lehmer's totient problem.

Contribution

The paper introduces a new Euler totient inequality, relates parameters to Lehmer numbers, and derives conditions limiting the occurrence of Lehmer numbers among primes and square-free numbers.

Findings

Established an Euler totient inequality.

Showed at most one of n or pn is a Lehmer number for primes p and odd square-free n.

Proposed open problems for future research on Lehmer numbers.

Abstract

A composite number $n$ is called a Lehmer number when $\phi(n) | n - 1$, where $\phi$ is the Euler totient function. Lehmer's totient problem asks if there exist any composite numbers $n$ such that $\phi(n)| n-1$? No such numbers are known. In this paper we establish an Euler Totient Inequality and relate some new parameters to Lehmer numbers. As an application of what we have done, we show that for all prime numbers $p$ and for all odd square-free numbers $n$, at most one of $n$ or $pn$ is a Lehmer number. Finally we suggest some open problems for the future investigations on the Lehmer numbers.