On numbers $n$ with polynomial image coprime with the $n$th term of a linear recurrence

TL;DR

This paper investigates the set of natural numbers where the gcd of a linear recurrence and a polynomial is a fixed integer, proving the set's natural density exists and characterizing when it is zero.

Contribution

It establishes the existence of natural density for sets defined by gcd conditions involving linear recurrences and polynomials, and characterizes when this density is zero.

Findings

The set of numbers with gcd equal to h has a natural density.

Under certain conditions, the density of coprime cases is zero iff the set is finite.

The results apply to non-degenerate recurrences and polynomials with no fixed divisors.

Abstract

Let $F$ be an integral linear recurrence, $G$ be an integer-valued polynomial splitting over the rationals, and $h$ be a positive integer. Also, let $\mathcal{A}_{F,G,h}$ be the set of all natural numbers $n$ such that $\gcd(F(n), G(n)) = h$. We prove that $\mathcal{A}_{F,G,h}$ has a natural density. Moreover, assuming $F$ is non-degenerate and $G$ has no fixed divisors, we show that $\mathbf{d}(\mathcal{A}_{F,G,1}) = 0$ if and only if $\mathcal{A}_{F,G,1}$ is finite.