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International Journal of Mathematics and Mathematical Sciences
Volume 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

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 be the th Fibonacci number. The order of appearance of a natural number is defined as the smallest natural number such that divides . For instance, for all , we have . In this paper, we will construct infinitely many natural numbers satisfying the previous inequalities and which do not belong to the Fibonacci sequence.

1. Introduction

Let be the Fibonacci sequence given by , for , where and . A few terms of this sequence are

The Fibonacci numbers are well known for possessing wonderful and amazing properties (consult [1] together with its very extensive annotated bibliography for additional references and history). In 1963, the Fibonacci Association was created to provide enthusiasts an opportunity to share ideas about these intriguing numbers and their applications. Also, in the issues of The Fibonacci Quarterly, we can find many new facts, applications, and relationships about Fibonacci numbers.

Let be a positive integer number, the order (or rank) of appearance of in the Fibonacci sequence, denoted by , is defined as the smallest positive integer , such that (some authors also call it order of apparition, or Fibonacci entry point). There are several results about in the literature. For instance, every positive integer divides some Fibonacci number, that is, for all . The proof of this fact is an immediate consequence of the Théorème Fondamental of Section XXVI in [2, page 300]. Also, it is a simple matter to prove that , for . In fact, if with , then divides , for some and thus with . Therefore, the inequality gives . So the order of appearance of a Fibonacci number is locally smallest in this sense. On the other hand, there are integers for which is locally smallest but which are not Fibonacci numbers, for example, So, a natural question arises: are there infinitely many natural numbers that do not belong to the Fibonacci sequence and such that ?

In this note, we give an affirmative answer to this question by proving the following.

Theorem 1.1. Given an integer , the number has order of appearance , for all . In particular, it is not a Fibonacci number. Moreover, one has for all sufficiently large .

2. Proof of Theorem 1.1

We recall that the problem of the existence of infinitely many prime numbers in the Fibonacci sequence remains open; however, several results on the prime factors of a Fibonacci number are known. For instance, a primitive divisor of is a prime factor of that does not divide . In particular, . It is known that a primitive divisor of exists whenever . The above statement is usually referred to the Primitive Divisor Theorem (see [3] for the most general version).

Now, we are ready to deal with the proof of the theorem.

Since divides , then . On the other hand, if divides , then we get the relation where is a positive integer number. Since , the Primitive Divisor Theorem implies that . Therefore, yielding . Now, if is a Fibonacci number, say , we get which leads to an absurdity as (keep in mind that ). Therefore, is not a Fibonacci number, for all .

Now, it suffices to prove that , or equivalently, if divides , then , for all sufficiently large , where .

Let be a positive integer number such that . If , we have where in the last inequality above, we used the fact that , for . Thus, as desired. For finishing the proof, it suffices to show that there exist only finitely many pairs of positive integers, such that or equivalently,

Towards a contradiction, suppose that (2.4) have infinitely many solutions with and . Hence, and are unbounded sequences. Since is bounded, we can assume, without loss of generality, that is a constant, say , for all sufficiently large (by the reordering of indexes if necessary). Now, by (2.4), we get On the other hand, the well-known Binet's formula leads to Thus,

Combining (2.5) and (2.8), we get

Since is an integer and , we have that must be constant with respect to , say , for all sufficiently large. Therefore, (2.9) yields the relation and so (because is irrational for all nonzero rational number). But, this leads to (by (2.4)) which is absurd. This completes the proof of the theorem.

Acknowledgments

The author would like to express his gratitude to the anonymous referees for carefully examining this paper and providing a number of important comments, critics, and suggestions. One of their suggestions leads us to Theorem 1.1. The author also thanks FEMAT and CNPq for the financial support.

References

  1. T. Koshy, Fibonacci and Lucas Numbers with Applications, Pure and Applied Mathematics (New York), Wiley-Interscience, New York, NY, USA, 2001.
  2. E. Lucas, “Theorie des fonctions numeriques simplement periodiques,” American Journal of Mathematics, vol. 1, no. 4, pp. 289–321, 1878. View at Publisher · View at Google Scholar
  3. Yu. Bilu, G. Hanrot, and P. M. Voutier, “Existence of primitive divisors of Lucas and Lehmer numbers,” Journal für die Reine und Angewandte Mathematik, vol. 539, pp. 75–122, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH