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

Abstract

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 $(u_k{)}_{k\ge 0}$ be an integer sequence satisfying $u_k=a_1{u}_{k-1}+a_2{u}_{k-2}$, for every integer $k\ge 2$, where $u_0=0$, $u_1\ne 0$, and $a_1,a_2$ are fixed nonzero integers; and let $\alpha$ and $\beta$, with $|\alpha |\ge |\beta |$, 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 $$\log \mathrm{lcm}(u_a:a\in A)\sim \frac{\delta {\mathrm{Li}}_2(1-\delta )}{1-\delta }\cdot \frac{3\log |\alpha /\sqrt{(a_1^2,a_2)}|}{{\pi }^2}\cdot n^2$$ with probability $1-o(1)$, where $\mathrm{lcm}$ denotes the lowest common multiple, ${\mathrm{Li}}_2$ 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 Mátyás, who studied the deterministic case $\delta =1$, and is motivated by an asymptotic formula for $\mathrm{lcm}(A)$ due to Cilleruelo, Rué, Šarka, and Zumalacárregui.

References in corpus (1)

• A note on product sets of random sets

Cited by in corpus (1)

• On the least common multiple of shifted powers