# The density of numbers $n$ having a prescribed G.C.D. with the $n$th Fibonacci number

## TL;DR

This paper investigates the asymptotic density of positive integers n for which the gcd of n and the nth Fibonacci number is a fixed k, establishing its existence and providing a formula involving the Möbius function.

## Contribution

It proves the existence of the asymptotic density for sets defined by gcd conditions with Fibonacci numbers and provides an explicit formula for this density.

## Key findings

- Asymptotic density of _k exists for each k
- Explicit formula for the density involving the Möbius function and least common multiples
- Zero density occurs if and only if the set _k is empty

## Abstract

For each positive integer $k$, let $\mathscr{A}_k$ be the set of all positive integers $n$ such that $\gcd(n, F_n) = k$, where $F_n$ denotes the $n$th Fibonacci number. We prove that the asymptotic density of $\mathscr{A}_k$ exists and is equal to $$\sum_{d = 1}^\infty \frac{\mu(d)}{\operatorname{lcm}(dk, z(dk))}$$ where $\mu$ is the M\"obius function and $z(m)$ denotes the least positive integer $n$ such that $m$ divides $F_n$. We also give an effective criterion to establish when the asymptotic density of $\mathscr{A}_k$ is zero and we show that this is the case if and only if $\mathscr{A}_k$ is empty.