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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  