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DIVISIBILITY AMONG POWER GCD MATRICES AND POWER LCM MATRICES

Part of: Special matrices Elementary number theory Polynomials and matrices

Published online by Cambridge University Press:  01 September 2025

SHAOFANG HONG
SHAOFANG HONG*
Affiliation:
Mathematical College, Sichuan University , Chengdu 610064, PR China
*
e-mail: sfhong@scu.edu.cn
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Abstract

For any integers x and y, let $(x, y)$ and $[x, y]$ stand for the greatest common divisor and the least common multiple of x and y, respectively. Let $a,b$ and n be positive integers, and let $S=\{x_1, \ldots , x_n\}$ be a set of n distinct positive integers. We denote by $(S^a)$ and $[S^a]$ the $n\times n$ matrices having the ath power of $(x_i,x_j)$ and $[x_i,x_j]$, respectively, as the $(i,j)$-entry. Bourque and Ligh [‘On GCD and LCM matrices’, Linear Algebra Appl. 174 (1992), 65–74] showed that if S is factor closed (that is, S contains all positive divisors of any element of S), then the GCD matrix $(S)$ divides the LCM matrix $[S]$ (written as $(S)\mid [S]$) in the ring $M_n({\mathbb Z})$ of $n\times n$ matrices over the integers. Hong [‘Divisibility properties of power GCD matrices and power LCM matrices’, Linear Algebra Appl. 428 (2008), 1001–1008] proved that $(S^a)\mid (S^b)$, $(S^a)\mid [S^b]$ and $[S^a]\mid [S^b]$ in the ring $M_{n}({\mathbb Z})$ when $a\mid b$ and S is a divisor chain (namely, there is a permutation $\sigma $ of order n such that $x_{\sigma (1)}\mid \cdots \mid x_{\sigma (n)}$). In this paper, we show that if $a\mid b$ and S is factor closed, then $(S^a)\mid (S^b)$, $(S^a)\mid [S^b]$ and $[S^a]\mid [S^b]$ in the ring $M_{n}({\mathbb Z})$. The proof is algebraic and p-adic. Our result extends the Bourque–Ligh theorem. Finally, several interesting conjectures are proposed.

Keywords

power GCD matrixpower LCM matrixfactor-closed setdivisibilityp-adic valuation

MSC classification

Primary: 11C20: Matrices, determinants
Secondary: 11A25: Arithmetic functions; related numbers; inversion formulas 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors 15B36: Matrices of integers

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 113 , Issue 2 , April 2026 , pp. 231 - 243
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The research was partially supported by the National Science Foundation of China, Grant no. 12171332.

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