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, R0 and R1 be fixed integers, and let R=(Rn)n≥0 be the recurrence sequence defined by Rn+2=PRn+1−QRn for all n≥0. Under some conditions on the parameters, we determine a rational nontrivial divisor for Lk,n:= lcm(Rk,Rk+1,…,Rn), for all positive integers n and k, such that n≥k. As consequences, we derive nontrivial effective lower bounds for Lk,n, and we establish an asymptotic formula for log(Ln,n+m), where m is a fixed positive integer. Denoting by (Fn)n the usual Fibonacci sequence, we prove, for example, that for any m≥1, we have loglcm(Fn,Fn+1,…,Fn+m)∼n(m+1)logΦ( as n→+∞), where Φ denotes the golden ratio. We conclude the paper with some interesting identities and properties regarding the least common multiple of Lucas sequences.