On The Lehmer Numbers, I

TL;DR

This paper investigates Lehmer numbers, a special class of composite numbers related to Euler's totient function, providing new necessary conditions and methods to construct numbers that are not Lehmer numbers, advancing understanding of this open problem.

Contribution

It introduces new necessary conditions for Lehmer numbers and proposes a method to construct numbers that are definitively not Lehmer numbers.

Findings

Lehmer numbers must be odd and square-free.

New necessary conditions for identifying Lehmer numbers.

A construction method for non-Lehmer numbers.

Abstract

A composite number $n$ is called Lehmer when $\phi(n) | n - 1$, where $\phi$ is the Euler totient function. In 1932, D.~H.~Lehmer conjectured that there are no composite Lehmer numbers and showed that Lehmer numbers must be odd and square-free. Although a number of additional constraints have been found since, the problem remains still open. For each odd number $m>1$, let $m^\star$ be the largest number such that $2^{m^\star}$ divides $m-1$. Using this notion we present some new necessary conditions and introduce a method to construct some new family of numbers $n$ which are not Lehmer number.