On the l.c.m. of shifted Fibonacci numbers

Carlo Sanna

Published 2020-07-27Version 1

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 \alpha}{\pi^2} \cdot n^2 \quad \text{as } n \to +\infty, \end{equation*} where $\operatorname{lcm}$ is the least common multiple and $\alpha := \big(1 + \sqrt{5}) / 2$ 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 \alpha}{\pi^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 \alpha}{\pi^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.