On the least common multiple of binary linear recurrence sequences (2011.03858v1)

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.