论文标题
在L.C.M.斐波那契数的变化
On the l.c.m. of shifted Fibonacci numbers
论文作者
论文摘要
令$（f_n）_ {n \ geq 1} $为fibonacci编号的序列。 Guy and Matiyasevich证明了\ begin {qore*} \ log \ protatorName {lcm}（f_1，f_1，f_2，\ dots，f_n）\ sim \ sim \ sim \ sim \ frac {3 \logα} \ end {equation*}其中$ \ operatatorName {lcm} $是最不常见的倍数，$α：= \ big（1 + \ sqrt {5}） / 2 $是黄金比率。 我们证明，对于每个周期序列$ \ MATHBF {s} =（s_n）_ {n \ geq 1} $ in $ \ { - 1，+1，+1 \} $，存在一个有效的可计算理性数量$ c _ {\ Mathbf {s}}}}} $ c _ { （f_3 + s_3，f_4 + s_4，\ dots，f_n + s_n）\ sim \ frac {3 \logα} {π^2} \ cdot c_ \ cdot c_ \ mathbf {s} \ cdot n^2，\ quad \ quad \ quad \ quad \ quad \ text {as} n \ text}此外，我们表明，如果$（s_n）_ {n \ geq 1} $是$ \ { - 1， + 1 \} $中的独立均匀分布的随机变量的顺序f_4 + s_4，\ dots，f_n + s_n）\ big] \ sim \ sim \ frac {3 \logα} {π^2} \ cdot \ cdot \ frac {15 \ propatatorname {li} _2（li} _2（1 /16）} \ end {equation*}其中$ \ operatatorName {li} _2 $是dilogarithm函数。
Let $(F_n)_{n \geq 1}$ be the sequence of Fibonacci numbers. Guy and Matiyasevich proved that \begin{equation*} \log \operatorname{lcm} (F_1, F_2, \dots, F_n) \sim \frac{3 \log α}{π^2} \cdot n^2 \quad \text{as } n \to +\infty, \end{equation*} where $\operatorname{lcm}$ is the least common multiple and $α:= \big(1 + \sqrt{5}) / 2$ is the golden ratio. We prove that for every periodic sequence $\mathbf{s} = (s_n)_{n \geq 1}$ in $\{-1,+1\}$ there exists an effectively computable rational number $C_{\mathbf{s}} > 0$ such that \begin{equation*} \log \operatorname{lcm} (F_3 + s_3, F_4 + s_4, \dots, F_n + s_n) \sim \frac{3 \log α}{π^2} \cdot C_\mathbf{s} \cdot n^2 , \quad \text{as } n \to +\infty . \end{equation*} Moreover, we show that if $(s_n)_{n \geq 1}$ is a sequence of independent uniformly distributed random variables in $\{-1,+1\}$ then \begin{equation*} \mathbb{E}\big[\log \operatorname{lcm} (F_3 + s_3, F_4 + s_4, \dots, F_n + s_n)\big] \sim \frac{3 \log α}{π^2} \cdot \frac{15 \operatorname{Li}_2(1 / 16)}{2} \cdot n^2 , \quad \text{as } n \to +\infty , \end{equation*} where $\operatorname{Li}_2$ is the dilogarithm function.
