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.
Categories: math.NT