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ON NUMBERS $n$ WITH POLYNOMIAL IMAGE COPRIME WITH THE $n$TH TERM OF A LINEAR RECURRENCE

Part of: Elementary number theory Sequences and sets Multiplicative number theory

Published online by Cambridge University Press:  28 August 2018

DANIELE MASTROSTEFANO Open the ORCID record for DANIELE MASTROSTEFANO [Opens in a new window]  and
CARLO SANNA Open the ORCID record for CARLO SANNA [Opens in a new window]
DANIELE MASTROSTEFANO
Affiliation:
Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, UK email danymastro93@hotmail.it
CARLO SANNA*
Affiliation:
Department of Mathematics, Università degli Studi di Torino, Turin, Italy email carlo.sanna.dev@gmail.com
*
✉carlo.sanna.dev@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

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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.

Keywords

greatest common divisorlinear recurrencesnatural densities

MSC classification

Primary: 11B37: Recurrences
Secondary: 11A07: Congruences; primitive roots; residue systems 11B39: Fibonacci and Lucas numbers and polynomials and generalizations 11N25: Distribution of integers with specified multiplicative constraints
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 1 , February 2019 , pp. 23 - 33
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The second author is a member of INdAM group GNSAGA.

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