On The Lehmer Numbers, I

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.