Open Access
2018 On the greatest common divisor of $n$ and the $n$th Fibonacci number
Paolo Leonetti, Carlo Sanna
Rocky Mountain J. Math. 48(4): 1191-1199 (2018). DOI: 10.1216/RMJ-2018-48-4-1191


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 $F_k$.


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Paolo Leonetti. Carlo Sanna. "On the greatest common divisor of $n$ and the $n$th Fibonacci number." Rocky Mountain J. Math. 48 (4) 1191 - 1199, 2018.


Published: 2018
First available in Project Euclid: 30 September 2018

zbMATH: 06958775
MathSciNet: MR3859754
Digital Object Identifier: 10.1216/RMJ-2018-48-4-1191

Primary: 11B39
Secondary: 11A05 , 11N25

Keywords: Fibonacci numbers , greatest common divisor , natural density , rank of appearance

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

Vol.48 • No. 4 • 2018
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