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arXiv · 1805.05114

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

Daniele Mastrostefano·Carlo Sanna
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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.

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Daniele Mastrostefano, Carlo Sanna. 2018-05-14. On numbers $n$ with polynomial image coprime with the $n$th term of a linear recurrence. https://doi.org/10.1017/s0004972718000606

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