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 $\#\left(\mathcal{A}\cap [1,x]\right)\gg x/\log x$ for all $x\ge 2$, 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 $F_k$.

References in corpus (2)

• On numbers $n$ relatively prime to the $n$th term of a linear recurrence
• Distribution of integral values for the ratio of two linear recurrences

Cited by in corpus (3)

• The density of numbers $n$ having a prescribed G.C.D. with the $n$th Fibonacci number
• On numbers $n$ relatively prime to the $n$th term of a linear recurrence
• Greatest common divisors of shifted primes and Fibonacci numbers