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On the greatest common divisor of n and the nth Fibonacci number, II

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

Published online by Cambridge University Press:  06 October 2022

Abhishek Jha Open the ORCID record for Abhishek Jha [Opens in a new window]  and
Carlo Sanna Open the ORCID record for Carlo Sanna [Opens in a new window]
Abhishek Jha
Affiliation:
Indraprastha Institute of Information Technology, Okhla Industrial Estate, Phase-3, New Delhi, India e-mail: abhishek20553@iiitd.ac.in
Carlo Sanna*
Affiliation:
Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino, Italy
*
e-mail: carlo.sanna@polito.it
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Abstract

Let $\mathcal {A}$ be the set of all integers of the form $\gcd (n, F_n)$ , where n is a positive integer and $F_n$ denotes the nth Fibonacci number. Leonetti and Sanna proved that $\mathcal {A}$ has natural density equal to zero, and asked for a more precise upper bound. We prove that

$$ \begin{align*} \#\big(\mathcal{A} \cap [1, x]\big) \ll \frac{x \log \log \log x}{\log \log x} \end{align*} $$
for all sufficiently large x. In fact, we prove that a similar bound also holds when the sequence of Fibonacci numbers is replaced by a general nondegenerate Lucas sequence.

Keywords

Fibonacci numbersgreatest common divisorLucas sequencerank of appearanceupper bound

MSC classification

Primary: 11B39: Fibonacci and Lucas numbers and polynomials and generalizations
Secondary: 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors 11N25: Distribution of integers with specified multiplicative constraints
Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 2 , June 2023 , pp. 617 - 625
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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