On the local behavior of the order of appearance in the Fibonacci sequence. (English) Zbl 1387.11016

Luca, Florian; Pomerance, Carl

Int. J. Number Theory 10, No. 4, 915-933 (2014).

The order of appearance \(z(N)\) in the Fibonacci sequence is the minimal positive integer \(k\) such that \(N\) divides \(F_k\), the \(k\)th Fibonacci number. The authors prove that all of the six possible orderings of \[ z(N), \; z(N+1), \; z(N+2) \] appear an infinite number of times as \(N=1,2,\ldots\) They conjecture that this holds true for longer patterns, too. Furthermore, they study the set \[ \mathcal{N}_c=\{N:\;\; z(N)=z(N+c)\}, \] where \(c\) is a nonzero integer. It is shown that for each even integer \(c\) and for each odd integer \(c\) with \(3\nmid z(c)\) the set \(\mathcal{N}_c\) is infinite and that for all \(c\) one has \[ \# \mathcal{N}_c(x)=O_c\left(\frac{x}{(\log x)^2}\right) \] for its counting function. The proofs use Baker’s theory for linear forms in logarithms of algebraic numbers, a result of Corvaja and Zannier on the height of rational functions at \(\mathcal{S}\)-unit points, the primitive divisor theorem, estimates on smooth numbers and sieve theory. In the final section, the authors give some heuristics that suggest that \(\#\mathcal{N}_c(x)\) is indeed of order \(x/(\log x)^2\).

Reviewer: Thomas Stoll (Vandœuvre-lés Nancy)

MSC:

11B39	Fibonacci and Lucas numbers and polynomials and generalizations
11D75	Diophantine inequalities
11N36	Applications of sieve methods

Keywords:

Fibonacci numbers; applications of the subspace theorem; applications of sieve methods

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