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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 \(F_k\).

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
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