International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 407643, 4 pages
http://dx.doi.org/10.1155/2011/407643
Research Article

On Integer Numbers with Locally Smallest Order of Appearance in the Fibonacci Sequence

Diego Marques

Departament of Mathematics, University of Brasilia, Brasilia-DF 70910-900, Brazil

Received 13 December 2010; Revised 7 February 2011; Accepted 27 February 2011

Academic Editor: Ilya M. Spitkovsky

Copyright © 2011 Diego Marques. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝐹 𝑛 be the 𝑛 th Fibonacci number. The order of appearance 𝑧 ( 𝑛 ) of a natural number 𝑛 is defined as the smallest natural number 𝑘 such that 𝑛 divides 𝐹 𝑘 . For instance, for all 𝑛 = 𝐹 𝑚 ≥ 5 , we have 𝑧 ( 𝑛 − 1 ) > 𝑧 ( 𝑛 ) < 𝑧 ( 𝑛 + 1 ) . In this paper, we will construct infinitely many natural numbers satisfying the previous inequalities and which do not belong to the Fibonacci sequence.