On the Euler Function of Fibonacci Numbers

Florian Luca
Instituto de Matemáticas
Universidad Nacional Autonoma de México
C.P. 58089, Morelia, Michoacán
México
mailto:fluca@matmor.unam.mxfluca@matmor.unam.mx

V. Janitzio Mejía Huguet
Departamento de Ciencias Básicas
Universidad Autónoma Metropolitana-Azcapotzalco
Av. San Pablo #180
Col. Reynosa Tamaulipas
C. P. 02200, Azcapotzalco DF
México
mailto:vjanitzio@gmail.comvjanitzio@gmail.com

Florin Nicolae
Institut für Mathematik
Technische Universität Berlin
MA 8-1, Strasse des 17 Juni 136
D-10623 Berlin
Germany
mailto:nicolae@math.tu-berlin.denicolae@math.tu-berlin.de
and
Institute of Mathematics
Romanian Academy
P.O. Box 1-764, RO-014700, Bucharest
Romania

in

Abstract:

In this paper, we show that for any positive integer $k$, the set

$$\left\{\left(\frac{\phi(F_{n+1})}{\phi(F_n)},\frac{\phi(F_{n+2})}{\phi(F_n)},\ldots,\frac{\phi(F_{n+k})}{\phi(F_n)}\right):n\ge 1\right\}$$

is dense in $\mathbb{R}^{k}_{\ge 0}$, where $\phi(m)$ is the Euler function of the positive integer $m$ and $F_n$ is the $n$th Fibonacci number.