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On the greatest common divisor of nn and the nnth Fibonacci number

Abstract

Let A\mathcal{A} be the set of all integers of the form gcd(n,Fn)\gcd(n, F_n), where nn is a positive integer and FnF_n denotes the nnth Fibonacci number. We prove that #(A[1,x])x/logx\#\left(\mathcal{A} \cap [1, x]\right) \gg x / \log x for all x2x \geq 2, and that A\mathcal{A} has zero asymptotic density. Our proofs rely on a recent result of Cubre and Rouse which gives, for each positive integer nn, an explicit formula for the density of primes pp such that nn divides the rank of appearance of pp, that is, the smallest positive integer kk such that pp divides FkF_k

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