Factorization of power GCD matrices and power LCM matrices on certain gcd-closed sets

TL;DR

The paper addresses divisibility relations among $a$-th power GCD matrices $(S^a)$ and $a$-th power LCM matrices $[S^a]$ on gcd-closed sets, focusing on the regime where $a|b$ and the maximum number of greatest-type divisors satisfies $igl|G_S(x)igr|=3$ under a condition $\oldsymbol{\mathcal{G}}$. It develops a framework of auxiliary sums and inverse formulas, proving the integrality of expressions that certify $(S^a) mid (S^b)$, $(S^a) mid [S^b]$, and $[S^a] mid [S^b]$ in $M_{|S|}(\mathbb{Z})$, thus extending the Chen–Hong–Zhao result and partially confirming Hong's conjecture for the case $|G_S(x)|=3$. The work relies on detailed case analyses of greatest-type divisors and leverages the condition $oldsymbol{\mathcal{G}}$ to control gcd/LCM relationships, ultimately contributing to the structural understanding of divisibility among structured arithmetic matrices. The findings may impact algebraic determinations of determinant and invertibility properties of power GCD/LCM matrices in number theory and related combinatorial settings.

Abstract

For integers $x$ and $y$, $(x, y)$ and $[x, y]$ stand for the greatest common divisor and the least common multiple of $x$ and $y$ respectively. Denote by $|T|$ the number of elements of a finite set $T$. Let $a,b$ and $n$ be positive integers and let $S=\{x_1, \cdots, x_n\}$ be a set of $n$ distinct positive integers. We denote by $(S^a)$ (resp. $[S^a]$) the $n\times n$ matrix having the $a$th power of $(x_i,x_j)$ (resp. $[x_i,x_j]$) as its $(i,j)$-entry. For any $x\in S$, define $G_{S}(x):=\{d\in S: d<x, d|x \ {\rm and} \ (d|y|x, y\in S) \Rightarrow y\in \{d,x\}\}$. In this paper, we show that if $a|b$ and $S$ is gcd closed (namely, $(x_i, x_j)\in S$ for all integers $i$ and $j$ with $1\le i, j\le n$) and $\max_{x\in S}\{|G_S (x)|\}=3$ such that any elements $y_1,y_2\in G_S(x)$ satisfy that $[y_1,y_2]=x$ and $(y_1,y_2)\in G_S(y_1)\cap G_S(y_2)$), then $(S^a)|(S^b)$, $(S^a)|[S^b]$ and $[S^a]|[S^b]$ hold in the ring $M_{n}({\mathbb Z})$. This extends the Chen-Hong-Zhao theorem gotten in 2022. This also partially confirms a conjecture of Hong raised in [S.F. Hong, Divisibility among power GCD matrices and power LCM matrices, {\it Bull. Aust. Math. Soc.}, doi:10.1017/S0004972725100361].

Theorems & Definitions (24)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6