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

TL;DR

This paper proves Hong's conjecture that power GCD and LCM matrices on gcd-closed sets with a specific condition are divisible by lower power GCD matrices.

Contribution

It confirms Hong's conjecture by establishing divisibility properties of power GCD and LCM matrices under certain conditions.

Findings

Power GCD and LCM matrices are divisible by lower power GCD matrices on gcd-closed sets.

The divisibility holds for arbitrary positive integers a and b with a dividing b.

The result confirms a conjecture proposed by Hong in 2026.

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].