On the least common multiple of binary linear recurrence 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 $\mathbf{R}={\left(R_n\right)}_n$ be the recurrence sequence defined by $R_{n+2}=P{R}_{n+1}-Q{R}_n$ $(\forall n\ge 0)$. 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 $n\ge k$. 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 \[\log \mathrm{lcm}\left(F_{n},F_{n+1},\dots,F_{n+m}\right)\sim n(m+1)\logΦ~~~~\text{as}~n\rightarrow +\infty,\] where $\Phi$ denotes the golden ratio. We conclude the paper by some interesting identities and properties regarding the least common multiple of Lucas sequences.

18 pages