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ON THE DIVISIBILITY AMONG POWER LCM MATRICES ON GCD-CLOSED SETS

Published online by Cambridge University Press:  19 May 2022

GUANGYAN ZHU*
Affiliation:
Mathematical College, Sichuan University, Chengdu610064, PR China and School of Teacher Education, Hubei Minzu University, Enshi445000, PR China
MAO LI
Affiliation:
School of Mathematics and Statistics, Southwest University, Chongqing400715, PR China e-mail: limao@swu.edu.cn

Abstract

Let $a,b$ and n be positive integers and let $S=\{x_1, \ldots , x_n\}$ be a set of n distinct positive integers. For ${x\in S}$ , define $G_{S}(x)=\{d\in S: d<x, \,d\mid x \ \mathrm {and} \ (d\mid y\mid x, y\in S)\Rightarrow y\in \{d,x\}\}$ . Denote by $[S^a]$ the $n\times n$ matrix having the ath power of the least common multiple of $x_i$ and $x_j$ as its $(i,j)$ -entry. We show that the bth power matrix $[S^b]$ is divisible by the ath power matrix $[S^a]$ if $a\mid b$ and S is gcd closed (that is, $\gcd (x_i, x_j)\in S$ for all integers i and j with $1\le i, j\le n$ ) and $\max _{x\in S} \{|G_S (x)|\}=1$ . This confirms a conjecture of Shaofang Hong [‘Divisibility properties of power GCD matrices and power LCM matrices’, Linear Algebra Appl.428 (2008), 1001–1008].

Type
Research Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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ON THE DIVISIBILITY AMONG POWER LCM MATRICES ON GCD-CLOSED SETS
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