The Euler Totient Function on Lucas Sequences

Abstract

In 2009, Luca and Nicolae proved that the only Fibonacci numbers whose Euler totient function is another Fibonacci number are $1,2$, and $3$. In 2015, Faye and Luca proved that the only Pell numbers whose Euler totient function is another Pell number are $1$ and $2$. Here we add to these two results and prove that for any fixed natural number $P\ge 3$, if we define the sequence ${\left(u_n\right)}_n$ as $u_0=0$, $u_1=1$, and $u_n=P{u}_{n-1}+u_{n-2}$ for all $n\ge 2$, then the only solution to the Diophantine equation $\phi \left(u_n\right)=u_m$ is $\phi \left(u_1\right)=\phi (1)=1=u_1$.