A new upper bound for numbers with the Lehmer property and its application to repunit numbers

Abstract

A composite positive integer $n$ has the Lehmer property if $\phi(n)$ divides $n-1,$ where $\phi$ is an Euler totient function. In this note we shall prove that if $n$ has the Lehmer property, then $n\leq 2^{2^{K}}-2^{2^{K-1}}$, where $K$ is the number of prime divisors of $n$. We apply this bound to repunit numbers and prove that there are at most finitely many numbers with the Lehmer property in the set $$ \left\{\frac{g^{n}-1}{g-1}\ \bigg|\ n,g\in\mathbb{N},\ \nu_{2}(g)+\nu_{2}(g+1)\leq L\ \right\}, $$ where $\nu_{2}(g)$ denotes the highest power of $2$ that divides $g$, and $L\geq 1$ is a fixed real number.