On the least common multiple of binary linear recurrence sequences

TL;DR

This paper develops a method to estimate the least common multiple of binary linear recurrence sequences, providing bounds, asymptotic formulas, and properties, with applications to Fibonacci and Lucas sequences.

Contribution

It introduces a new approach for estimating the LCM of binary linear recurrence sequences, including explicit divisors, bounds, and asymptotic behavior, extending understanding of these sequences.

Findings

Established a rational divisor for the LCM of recurrence sequences.

Derived effective lower bounds for the LCM.

Proved asymptotic formulas for the logarithm of the LCM of Fibonacci sequences.

Abstract

In this paper, 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 $\boldsymbol{R}=\left(R_n\right)_{n}$ be the recurrence sequence defined by $R_{n+2}=PR_{n+1}-QR_{n}$ $(\forall n\geq 0)$. Under some conditions on the parameters, we determine a rational nontrivial divisor for $L_{k,n}:=\mathrm{lcm}\left(R_k,R_{k+1},\dots,R_n\right)$, for all positive integers $n$ and $k$, such that $n\geq k$. 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 $m\geq 1$, we have \[\log \mathrm{lcm}\left(F_{n},F_{n+1},\dots,F_{n+m}\right)\sim n(m+1)\log\Phi~~~~\text{as}~n\rightarrow +\infty,\] where $\Phi$ denotes the golden ratio. We conclude the paper by some interesting identities and properties regarding the least common multiple of Lucas sequences.