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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, R0 and R1 be fixed integers, and let R=(Rn)n0 be the recurrence sequence defined by Rn+2=PRn+1QRn for all n0. Under some conditions on the parameters, we determine a rational nontrivial divisor for Lk,n:= lcm(Rk,Rk+1,,Rn), for all positive integers n and k, such that nk. As consequences, we derive nontrivial effective lower bounds for Lk,n, and we establish an asymptotic formula for log(Ln,n+m), where m is a fixed positive integer. Denoting by (Fn)n the usual Fibonacci sequence, we prove, for example, that for any m1, we have

loglcm(Fn,Fn+1,,Fn+m)n(m+1)logΦ( as n+),

where Φ denotes the golden ratio. We conclude the paper with some interesting identities and properties regarding the least common multiple of Lucas sequences.

Citation

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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

Rights: Copyright © 2021 Rocky Mountain Mathematics Consortium

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Vol.51 • No. 5 • October 2021
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