论文标题
在二进制线性复发序列的最少常见倍数上
On the least common multiple of binary linear recurrence sequences
论文作者
论文摘要
在本文中，我们提出了一种估计一类二进制线性复发序列中最小常见倍数的方法。令$ p，q，r_0 $和$ r_1 $为整数固定，然后让$ \ boldsymbol {r} = \ left（r_n \ right）_ {n} $为$ r_ {n+2} = pr_ {n+1} -qr_ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n} $ n}。在某些条件下，我们确定了$ l_ {k，n}的合理非平凡除数：= \ m atrm {lcm} \ left（r_k，r_k，r_ {k+1}，\ dots，r_n \ right）$，用于所有积极的integers $ n $ n $ n $ n $ and $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n。结果，我们为$ l_ {k，n} $得出了非平凡有效的下限，并为$ \ log \ left建立了一个渐近公式（l_ {n，n+m} \ right）$，其中$ m $是固定的阳性integer。按$ \ left（f_n \右）表示_ {n} $通常的fibonacci序列，例如，我们证明，对于任何$ m \ geq 1 $，我们都有\ [\ log \ log \ m mathrm {lcm} \ left（f_ {n}，f_ {n}，f_ {n+1}，f_ { n（m +1）\logφ~~~~ \ text {as} 〜n \ rightarrow +\ infty，\]其中$φ$表示黄金比率。我们以一些有趣的身份和属性结束了有关Lucas序列最不常见倍数的特性。
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 \[\log \mathrm{lcm}\left(F_{n},F_{n+1},\dots,F_{n+m}\right)\sim n(m+1)\logΦ~~~~\text{as}~n\rightarrow +\infty,\] where $Φ$ denotes the golden ratio. We conclude the paper by some interesting identities and properties regarding the least common multiple of Lucas sequences.
