Translator Disclaimer
October 2021 On the least common multiple of binary linear recurrence sequences
Sid Ali Bousla
Rocky Mountain J. Math. 51(5): 1583-1597 (October 2021). DOI: 10.1216/rmj.2021.51.1583

Abstract

We present a method for estimating the least common multiple of a large class of binary linear recurrence sequences. Let $P$, $Q$, ${R_0}$ and ${R_1}$ be fixed integers, and let $R = {\left( {{R_n}} \right)_{n \ge 0}}$ be the recurrence sequence defined by ${R_{n + 2}} = P{R_{n + 1}} - Q{R_n}$ for all $n \ge 0$. Under some conditions on the parameters, we determine a rational nontrivial divisor for ${L_{k,n}}: = lcm\left( {{R_k},{R_{k + 1}}, \ldots ,{R_n}} \right)$, for all positive integers $n$ and $k$, such that $n \ge k$. As consequences, we derive nontrivial effective lower bounds for ${L_{k,n}}$, and we establish an asymptotic formula for $\log \left( {{L_{n,n + m}}} \right)$, where $m$ is a fixed positive integer. Denoting by ${\left( {{F_n}} \right)_n}$ the usual Fibonacci sequence, we prove, for example, that for any $m \ge 1$, we have

$\log lcm\left( {{F_n},{F_{n + 1}}, \ldots ,{F_{n + m}}} \right) \sim n\left( {m + 1} \right)\log {\rm{\Phi }}\quad \left( {{\rm{\;as\;}}n \to + \infty } \right),$

where ${\rm{\Phi }}$ denotes the golden ratio. We conclude the paper with some interesting identities and properties regarding the least common multiple of Lucas sequences.

Citation

Sid Ali Bousla. "On the least common multiple of binary linear recurrence sequences." Rocky Mountain J. Math. 51 (5) 1583 - 1597, October 2021. https://doi.org/10.1216/rmj.2021.51.1583

Information

Received: 21 November 2020; Revised: 27 February 2021; Accepted: 2 March 2021; Published: October 2021
First available in Project Euclid: 17 February 2022

Digital Object Identifier: 10.1216/rmj.2021.51.1583

Subjects:
Primary: 11A05 , 11B39 , 11B83
Secondary: 11B65

Keywords: Asymptotic formula , binary recurrence sequence , Fibonacci sequence , least common multiple , Lucas sequence

JOURNAL ARTICLE
15 PAGES

It is not available for individual sale.

SHARE
Vol.51 • No. 5 • October 2021