## 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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### References:

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