The Euler Totient Function on Lucas Sequences

TL;DR

This paper extends previous results on Fibonacci and Pell numbers, proving that for a broad class of Lucas sequences, the only case where the Euler totient of a sequence term equals another sequence term is trivial.

Contribution

It generalizes earlier findings by proving a universal property of Lucas sequences regarding the Euler totient function and sequence terms.

Findings

The only solution for the generalized Lucas sequence is when both terms are 1.

The result holds for any fixed P ≥ 3 in the recurrence.

No non-trivial solutions exist for the Diophantine equation φ(u_n)=u_m.

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 $P\geq 3$, if we define the sequence $\left(u_n\right)_n$ as $u_0=0$, $u_1=1$, and $u_n=Pu_{n-1}+u_{n-2}$ for all $n\geq 2$, then the only solution to the Diophantine equation $\varphi\left(u_n\right)=u_m$ is $\varphi\left(u_1\right)=\varphi(1)=1=u_1$.