On the greatest common divisor of $n$ and the $n$th Fibonacci number

Abstract

Let $\mathcal {A}$ be the set of all integers of the form $\gcd (n, F_n)$, 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 Cubre and Rouse 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 $F_k$.