Lucas Numbers with Lehmer Property

Abstract

A composite positive integer n is Lehmer if ϕ(n) divides n-1, where ϕ(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 $(L_n{)}_{n\ge 0}$ given by $L_0=2,L_1=1$ and $L_{n+2}=L_{n+1}+L_n$ for all $n\ge 0.$