Lucas Numbers with Lehmer Property

TL;DR

This paper proves that no Lehmer numbers exist within the Lucas sequence, extending previous results that showed their absence in the Fibonacci sequence, using adapted mathematical methods.

Contribution

It demonstrates the non-existence of Lehmer numbers in the Lucas sequence, building on prior work on Fibonacci numbers with a novel adaptation of existing methods.

Findings

No Lehmer number in Lucas sequence

Extension of non-existence results from Fibonacci to Lucas

Method adaptation for Lucas sequence

Abstract

A composite positive integer n is Lehmer if \phi(n) divides n-1, where \phi(n) is the Euler's totient function. No Lehmer number is known, nor has it been proved that they don't exist. In 2007, the second author [7] proved that there is no Lehmer number in the Fibonacci sequence. In this paper, we adapt the method from [7] to show that there is no Lehmer number in the companion Lucas sequence of the Fibonacci sequence $(L_n)_{n\geq 0}$ given by $L_0 = 2, L_1 = 1$ and $L_{n+2} = L_{n+1} + L_n$ for all $n\geq 0.$