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On the greatest common divisor of $$n$$ and the $$n$$th Fibonacci number. (English) Zbl 1437.11025
Summary: Let $$\mathcal{A}$$ be the set of all integers of the form $$\mathrm{gcd}(n,F_n)$$, where $$n$$ is a positive integer and $$F_n$$ denotes the $$n$$th Fibonacci number. We prove that $$\#(\mathcal{A}\cap [1,x])\gg x/\log x$$ for all $$x \geq 2$$ and that $$\mathcal{A}$$ has zero asymptotic density. Our proofs rely upon a recent result of P. Cubre and J. Rouse [Proc. Am. Math. Soc. 142, No. 11, 3771–3785 (2014; Zbl 1309.11012)] which gives, for each positive integer $$n$$, an explicit formula for the density of primes $$p$$ such that $$n$$ divides the rank of appearance of $$p$$, that is, the smallest positive integer $$k$$ such that $$p$$ divides $$F_k$$.

##### MSC:
 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors 11N25 Distribution of integers with specified multiplicative constraints
OEIS
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