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- Title
ON NUMBERS $n$ WITH POLYNOMIAL IMAGE COPRIME WITH THE $n$ TH TERM OF A LINEAR RECURRENCE.
- Authors
MASTROSTEFANO, DANIELE; SANNA, CARLO
- Abstract
Let $F$ be an integral linear recurrence, $G$ an integer-valued polynomial splitting over the rationals and $h$ 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 that $F$ is nondegenerate and $G$ has no fixed divisors, we show that the density of ${\mathcal{A}}_{F,G,1}$ is 0 if and only if ${\mathcal{A}}_{F,G,1}$ is finite.
- Subjects
POLYNOMIALS; NATURAL numbers; MOBIUS function; FIBONACCI sequence; LUCAS sequence
- Publication
Bulletin of the Australian Mathematical Society, 2019, Vol 99, Issue 1, p23
- ISSN
0004-9727
- Publication type
Article
- DOI
10.1017/S0004972718000606