On the l.c.m. of random terms of binary recurrence sequences

Carlo Sanna

Published 2019-09-10Version 1

For every positive integer $n$ and every $\delta \in [0,1]$, let $B(n, \delta)$ denote the probabilistic model in which a random set $A \subseteq \{1, \dots, n\}$ is constructed by choosing independently every element of $\{1, \dots, n\}$ with probability $\delta$. Moreover, let $(u_k)_{k \geq 0}$ be an integer sequence satisfying $u_k = a_1 u_{k - 1} + a_2 u_{k - 2}$, for every integer $k \geq 2$, where $u_0 = 0$, $u_1 \neq 0$, and $a_1, a_2$ are fixed nonzero integers; and let $\alpha$ and $\beta$, with $|\alpha| \geq |\beta|$, be the two roots of the polynomial $X^2 - a_1 X - a_2$. Also, assume that $\alpha / \beta$ is not a root of unity. We prove that, as $\delta n / \log n \to +\infty$, for every $A$ in $B(n, \delta)$ we have $$\log \operatorname{lcm} (u_a : a \in A) \sim \frac{\delta\operatorname{Li}_2(1 - \delta)}{1 - \delta} \cdot \frac{3\log\!\big|\alpha / \!\sqrt{(a_1^2, a_2)}\big|}{\pi^2} \cdot n^2 $$ with probability $1 - o(1)$, where $\operatorname{lcm}$ denotes the lowest common multiple, $\operatorname{Li}_2$ is the dilogarithm, and the factor involving $\delta$ is meant to be equal to $1$ when $\delta = 1$. This extends previous results of Akiyama, Tropak, Matiyasevich, Guy, Kiss and M\'aty\'as, who studied the deterministic case $\delta = 1$, and is motivated by an asymptotic formula for $\operatorname{lcm}(A)$ due to Cilleruelo, Ru\'{e}, \v{S}arka, and Zumalac\'{a}rregui.