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

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 \(x/(\log x)^2\).

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11D75 Diophantine inequalities
11N36 Applications of sieve methods
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[1] DOI: 10.1016/0022-314X(83)90002-1 · Zbl 0513.10043
[2] DOI: 10.2307/1967797 · JFM 44.0216.01
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[6] DOI: 10.1155/2011/407643 · Zbl 1314.11007
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