The density of numbers n having a prescribed G.C.D. with the n-th Fibonacci number

Autor: Sanna, Carlo, Tron, Emanuele
Přispěvatelé: Università degli studi di Torino (UNITO), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)
Jazyk: angličtina
Rok vydání: 2018
Předmět:
Zdroj: Indagationes Mathematicae
Indagationes Mathematicae, Elsevier, 2018, 29 (3), pp.972-980. ⟨10.1016/j.indag.2018.03.002⟩
ISSN: 0019-3577
DOI: 10.1016/j.indag.2018.03.002⟩
Popis: For each positive integer $k$, let $\mathscr{A}_k$ be the set of all positive integers $n$ such that $\gcd(n, F_n) = k$, where $F_n$ denotes the $n$th Fibonacci number. We prove that the asymptotic density of $\mathscr{A}_k$ exists and is equal to $$\sum_{d = 1}^\infty \frac{\mu(d)}{\operatorname{lcm}(dk, z(dk))}$$ where $\mu$ is the M\"obius function and $z(m)$ denotes the least positive integer $n$ such that $m$ divides $F_n$. We also give an effective criterion to establish when the asymptotic density of $\mathscr{A}_k$ is zero and we show that this is the case if and only if $\mathscr{A}_k$ is empty.
Databáze: OpenAIRE
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