Cullen Numbers with the Lehmer Property

TL;DR

This paper proves that no Cullen number possesses the Lehmer property, meaning no such number is composite with its Euler totient dividing one less than itself.

Contribution

It establishes a non-existence result for Cullen numbers with the Lehmer property, a problem previously unresolved.

Findings

No Cullen number has the Lehmer property.

The result narrows the search for Lehmer numbers.

Supports the conjecture that Lehmer numbers are rare or nonexistent.

Abstract

Here, we show that there is no positive integer $n$ such that the $n$th Cullen number $C_n=n2^n+1$ has the property that it is composite but $\phi(C_n)\mid C_n-1$.