Rocky Mountain Journal of Mathematics

On the greatest common divisor of $n$ and the $n$th Fibonacci number

Paolo Leonetti and Carlo Sanna

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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 $n$th Fibonacci number. We prove that $\#(\mathcal {A} \cap [1, x])\gg x / \log x$ for all $x \geq 2$ and that $\mathcal {A}$ has zero asymptotic density. Our proofs rely upon a recent result of Cubre and Rouse which gives, for each positive integer $n$, an explicit formula for the density of primes $p$ such that $n$ divides the rank of appearance of $p$, that is, the smallest positive integer $k$ such that $p$ divides $F_k$.

Article information

Source
Rocky Mountain J. Math., Volume 48, Number 4 (2018), 1191-1199.

Dates
First available in Project Euclid: 30 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1538272829

Digital Object Identifier
doi:10.1216/RMJ-2018-48-4-1191

Mathematical Reviews number (MathSciNet)
MR3859754

Zentralblatt MATH identifier
06958775

Subjects
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

Keywords
Fibonacci numbers rank of appearance greatest common divisor natural density

Citation

Leonetti, Paolo; Sanna, Carlo. On the greatest common divisor of $n$ and the $n$th Fibonacci number. Rocky Mountain J. Math. 48 (2018), no. 4, 1191--1199. doi:10.1216/RMJ-2018-48-4-1191. https://projecteuclid.org/euclid.rmjm/1538272829


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