Journal of Integer Sequences, Vol. 12 (2009), Article 09.6.6

On the Euler Function of Fibonacci Numbers


Florian Luca
Instituto de Matemáticas
Universidad Nacional Autonoma de México
C.P. 58089, Morelia, Michoacán
México

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

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

Abstract:

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

$\displaystyle \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.

Full version:  pdf,    dvi,    ps,    latex    

(Concerned with sequence A000045.)

Received May 20 2009; revised version received September 22 2009. Published in Journal of Integer Sequences, September 22 2009.

Return to Journal of Integer Sequences home page