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

TL;DR

This paper establishes a new upper bound for composite numbers with the Lehmer property based on their prime divisors and applies this to show finitely many such numbers exist within certain sets of repunit numbers.

Contribution

It proves a novel upper bound for Lehmer numbers and demonstrates the finiteness of Lehmer numbers within a specific class of repunit numbers.

Findings

Lehmer numbers satisfy $n \,\leq\, 2^{2^{K}} - 2^{2^{K-1}}$ where $K$ is the number of prime divisors.

There are finitely many Lehmer numbers in the set of certain bounded repunit numbers.

The bound helps restrict the search for Lehmer numbers in specialized numeric sets.

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.