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

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Volume 2011, Article ID 407643,4pages doi: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

Correspondence should be addressed to Diego Marques,diego@mat.unb.br

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

Academic Editor: Ilya M. Spitkovsky

Copyrightq2011 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.

LetFnbe thenth Fibonacci number. The order of appearanceznof a natural numbernis defined

as the smallest natural numberk such thatndividesFk. For instance, for alln Fm ≥ 5, we havezn−1> zn< zn1. 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

LetFn_n≥0be the Fibonacci sequence given byFn2 Fn1Fn, forn≥ 0, whereF0 0 and

F11. A few terms of this sequence are

0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584, . . .. 1.1

The Fibonacci numbers are well known for possessing wonderful and amazing

propertiesconsult1together 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.

Letnbe a positive integer number, the order (or rank) of appearance ofnin the Fibonacci

sequence, denoted byzn, is defined as the smallest positive integerk, such thatn|Fksome

authors also call it order of apparition, or Fibonacci entry point. There are several results about

znin the literature. For instance, every positive integer divides some Fibonacci number, that

is,zn<∞for alln≥1. The proof of this fact is an immediate consequence of the Th´eor`eme

Fondamental of Section XXVI in2, page 300. Also, it is a simple matter to prove thatzFn−

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2 International Journal of Mathematics and Mathematical Sciences

for somej≥5 and thusFjuFmwithu≥2. Therefore, the inequalityFj≥2Fm2 > Fm

giveszFm j> mzFm. So the order of appearance of a Fibonacci number is locally

smallest in this sense. On the other hand, there are integersnfor whichznis locally smallest

but which are not Fibonacci numbers, for example,n 11,17,24,26,29,36,38,41,44,48, . . ..

So, a natural question arises: are there infinitely many natural numbersnthat do not belong

to the Fibonacci sequence and such thatzn−1> zn< zn1?

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

Theorem 1.1. Given an integerk ≥3, the numberNm :Fmk/Fkhas order of appearancemk, for

allm≥5. In particular, it is not a Fibonacci number. Moreover, one has

zNm−1> zNm< zNm1, 1.2

for all suﬃciently largem.

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 divisorpofFnis a prime factor ofFnthat does not divide

_n−1

j1Fj. In particular,zp n. It is known that a primitive divisorpofFn exists whenever

n ≥ 13. The above statement is usually referred to the Primitive Divisor Theoremsee3for

the most general version.

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

SinceNmdividesFmk, thenzNm≤mk. On the other hand, ifNmdividesFj, then we

get the relation

FkFj tFmk, 2.1

wheretis a positive integer number. Sincemk ≥ 15, the Primitive Divisor Theorem implies

thatj ≥mk. Therefore,zNm≥mkyieldingzNm mk. Now, ifNmis a Fibonacci number,

sayFt, we gett zNm mkwhich leads to an absurdity as Fk 1keep in mind that

k≥3. Therefore,Nmis not a Fibonacci number, for allm≥5.

Now, it suﬃces to prove that zNm > mk zNm, or equivalently, ifNm ±1

dividesFj, thenj > mk, for all suﬃciently largem, where∈ {±1}.

Letube a positive integer number such thatFj uNm. Ifu≥Fk1, we have

Fj≥

1 1

Fk

FmkFk1> Fmk, 2.2

where in the last inequality above, we used the fact that Fmk > FmFk > Fk 1Fk, for

m > k≥3. Thus,j > mkas desired. For finishing the proof, it suﬃces to show that there exist

only finitely many pairsk, jof positive integers, such that

Fj

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or equivalently,

uFmk−FkFj −uFk. 2.4

Towards a contradiction, suppose that2.4have infinitely many solutionsun, mn, jn

with un ∈ {1, . . . , Fk} and n ≥ 1. Hence, mn_n andjn_n are unbounded sequences. Since

un_nis bounded, we can assume, without loss of generality, thatunis a constant, sayu, for

all suﬃciently largenby the reordering of indexes if necessary. Now, by2.4, we get

lim

n→ ∞

Fmnk

Fjn

Fk

u. 2.5

On the other hand, the well-known Binet’s formula

Fn α

n₋₋₁n_α−n

√

5 , whereα

1√5

2 , 2.6

leads to

Fmnk

Fjn

αmnk−jn−−1mnkα−mnk−jn

1−−1jn_α−2jn . 2.7

Thus,

lim

n→ ∞

Fmnk

Fjn

lim

n→ ∞α

mnk−jn. _2.8

Combining2.5and2.8, we get

lim

n→ ∞α

mnk−jn Fk

u. 2.9

Sincemnk−jnis an integer and|α|> 1, we have thatmnk−jnmust be constant with

respect ton, sayt, for allnsuﬃciently large. Therefore,2.9yields the relationαtFk/u∈

and so t 0 because αt is irrational for all nonzero rational number. But, this leads to

by2.4

F2

kuFkFkFmnk−FkFmnk0, 2.10

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4 International Journal of Mathematics and Mathematical Sciences

Acknowledgments

The author would like to express his gratitude to the anonymous referees for carefully exam-ining this paper and providing a number of important comments, critics, and suggestions.

One of their suggestions leads us toTheorem 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 MathematicsNew York, Wiley-Interscience, New York, NY, USA, 2001.

2 E. Lucas, “Theorie des fonctions numeriques simplement periodiques,” American Journal of

Mathemat-ics, vol. 1, no. 4, pp. 289–321, 1878.

3 Yu. Bilu, G. Hanrot, and P. M. Voutier, “Existence of primitive divisors of Lucas and Lehmer numbers,”

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

Research Article