On the least common multiple of binary linear recurrence sequences

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 $L_{k,n}:=lcm\left(R_k,R_{k+1},\dots ,R_n\right)$, 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

$$loglcm\left(F_n,F_{n+1},\dots ,F_{n+m}\right)\sim n\left(m+1\right)log\mathrm{\Phi }\left(\text{ as }n\to +\infty \right),$$

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