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

## TL;DR

This paper investigates the set of all gcds of positive integers and their corresponding Fibonacci numbers, establishing their density properties and providing explicit formulas for related prime distributions.

## Contribution

It proves that the set of gcds has zero asymptotic density and grows at least as fast as x / log x, using recent results on prime divisibility in Fibonacci sequences.

## Key findings

- The set of gcds has zero asymptotic density.
- The number of such gcds up to x is at least proportional to x / log x.
- Explicit formulas for prime divisibility related to Fibonacci numbers are utilized.

## 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 $\#\left(\mathcal{A} \cap [1, x]\right) \gg x / \log x$ for all $x \geq 2$, and that $\mathcal{A}$ has zero asymptotic density. Our proofs rely on 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 $F_k$.