1 results
ON NUMBERS
$n$ WITH POLYNOMIAL IMAGE COPRIME WITH THE 
$n$TH TERM OF A LINEAR RECURRENCE
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 99 / Issue 1 / February 2019
- Published online by Cambridge University Press:
- 28 August 2018, pp. 23-33
- Print publication:
- February 2019
-
- Article
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- You have access
- Export citation
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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.
