Proof of Hong's conjecture on divisibility among power GCD and power LCM matrices on gcd-closed sets

Abstract

Let $a$ and $n$ be positive integers and let $S=\{x_1, \cdots, x_n\}$ be a set of $n$ distinct positive integers. For $x\in S$, one defines $G_{S}(x)=\{d\in S: d<x, d|x \ {\rm and} \ (d|y|x, y\in S)\Rightarrow y\in \{d,x\}\}$. We denote by $(S^a)$ (resp. $[S^a]$) the $n\times n$ matrix having the $a$th power of the greatest common divisor (resp. the least common multiple) of $x_i$ and $x_j$ as its $(i,j)$-entry. In this paper, we show that for arbitrary positive integers $a$ and $b$ with $a|b$, the $b$th power GCD matrix $(S^b)$ and the $b$th power LCM matrix $[S^b]$ are both divisible by the $a$th power GCD matrix $(S^a)$ if $S$ is a gcd-closed (i.e. $\gcd(x_i, x_j)\in S$ for all integers $i$ and $j$ with $1\le i,j\le n$) set satisfying the condition $\mathcal G$ (i.e., for any element $x\in S$, either $G_S(x)$ contains at most one element, or $G_S(x)$ contains at least two elements and satisfies that $[y_1,y_2]=x$ as well as $(y_1,y_2)\in G_S(y_1)\cap G_S(y_2)$ for any $\{y_1,y_2\}\subseteq G_S(x)$). This confirms a conjecture of Hong proposed in [S.F. Hong, Divisibility among power GCD matrices and power LCM matrices, {\it Bull. Aust. Math. Soc.} {\bf 113} (2026), 231-243].