On the divisibility of power LCM matrices by power GCD matrices

Jian Rong Zhao; Shaofang Hong; Qunying Liao; Kar-Ping Shum

Czechoslovak Mathematical Journal (2007)

  • Volume: 57, Issue: 1, page 115-125
  • ISSN: 0011-4642

Abstract

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Let S = { x 1 , ⋯ , x n } be a set of n distinct positive integers and e ≥ 1 an integer. Denote the n × n power GCD (resp. power LCM) matrix on S having the e -th power of the greatest common divisor ( x i , x j ) (resp. the e -th power of the least common multiple [ x i , x j ] ) as the ( i , j ) -entry of the matrix by ( ( x i , x j ) e ) (resp. ( [ x i , x j ] e ) ) . We call the set S an odd gcd closed (resp. odd lcm closed) set if every element in S is an odd number and ( x i , x j ) ∈ S (resp. [ x i , x j ] ∈ S ) for all 1 ≤ i , j ≤ n . In studying the divisibility of the power LCM and power GCD matrices, Hong conjectured in 2004 that for any integer e ≥ 1 , the n × n power GCD matrix ( ( x i , x j ) e ) defined on an odd-gcd-closed (resp. odd-lcm-closed) set S divides the n × n power LCM matrix ( [ x i , x j ] e ) defined on S in the ring M n ( ℤ ) of n × n matrices over integers. In this paper, we use Hong’s method developed in his previous papers [J. Algebra 218 (1999) 216–228; 281 (2004) 1–14, Acta Arith. 111 (2004), 165–177 and J. Number Theory 113 (2005), 1–9] to investigate Hong’s conjectures. We show that the conjectures of Hong are true for n ≤ 3 but they are both not true for n ≥ 4 .

How to cite

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Zhao, Jian Rong, et al. "On the divisibility of power LCM matrices by power GCD matrices." Czechoslovak Mathematical Journal 57.1 (2007): 115-125. <http://eudml.org/doc/31117>.

@article{Zhao2007,
abstract = {Let $S=\lbrace x_1,\dots ,x_n\rbrace $ be a set of $n$ distinct positive integers and $e\ge 1$ an integer. Denote the $n\times n$ power GCD (resp. power LCM) matrix on $S$ having the $e$-th power of the greatest common divisor $(x_i,x_j)$ (resp. the $e$-th power of the least common multiple $[x_i,x_j]$) as the $(i,j)$-entry of the matrix by $((x_i, x_j)^e)$ (resp. $([x_i, x_j]^e))$. We call the set $S$ an odd gcd closed (resp. odd lcm closed) set if every element in $S$ is an odd number and $(x_i,x_j)\in S$ (resp. $[x_i, x_j]\in S$) for all $1\le i,j \le n$. In studying the divisibility of the power LCM and power GCD matrices, Hong conjectured in 2004 that for any integer $e\ge 1$, the $n\times n$ power GCD matrix $((x_i, x_j)^e)$ defined on an odd-gcd-closed (resp. odd-lcm-closed) set $S$ divides the $n\times n$ power LCM matrix $([x_i, x_j]^e)$ defined on $S$ in the ring $M_n(\{\mathbb \{Z\}\})$ of $n\times n$ matrices over integers. In this paper, we use Hong’s method developed in his previous papers [J. Algebra 218 (1999) 216–228; 281 (2004) 1–14, Acta Arith. 111 (2004), 165–177 and J. Number Theory 113 (2005), 1–9] to investigate Hong’s conjectures. We show that the conjectures of Hong are true for $n\le 3$ but they are both not true for $n\ge 4$.},
author = {Zhao, Jian Rong, Hong, Shaofang, Liao, Qunying, Shum, Kar-Ping},
journal = {Czechoslovak Mathematical Journal},
keywords = {GCD-closed set; LCM-closed set; greatest-type divisor; divisibility; GCD-closed set; LCM-closed set; greatest-type divisor; divisibility},
language = {eng},
number = {1},
pages = {115-125},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the divisibility of power LCM matrices by power GCD matrices},
url = {http://eudml.org/doc/31117},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Zhao, Jian Rong
AU - Hong, Shaofang
AU - Liao, Qunying
AU - Shum, Kar-Ping
TI - On the divisibility of power LCM matrices by power GCD matrices
JO - Czechoslovak Mathematical Journal
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 1
SP - 115
EP - 125
AB - Let $S=\lbrace x_1,\dots ,x_n\rbrace $ be a set of $n$ distinct positive integers and $e\ge 1$ an integer. Denote the $n\times n$ power GCD (resp. power LCM) matrix on $S$ having the $e$-th power of the greatest common divisor $(x_i,x_j)$ (resp. the $e$-th power of the least common multiple $[x_i,x_j]$) as the $(i,j)$-entry of the matrix by $((x_i, x_j)^e)$ (resp. $([x_i, x_j]^e))$. We call the set $S$ an odd gcd closed (resp. odd lcm closed) set if every element in $S$ is an odd number and $(x_i,x_j)\in S$ (resp. $[x_i, x_j]\in S$) for all $1\le i,j \le n$. In studying the divisibility of the power LCM and power GCD matrices, Hong conjectured in 2004 that for any integer $e\ge 1$, the $n\times n$ power GCD matrix $((x_i, x_j)^e)$ defined on an odd-gcd-closed (resp. odd-lcm-closed) set $S$ divides the $n\times n$ power LCM matrix $([x_i, x_j]^e)$ defined on $S$ in the ring $M_n({\mathbb {Z}})$ of $n\times n$ matrices over integers. In this paper, we use Hong’s method developed in his previous papers [J. Algebra 218 (1999) 216–228; 281 (2004) 1–14, Acta Arith. 111 (2004), 165–177 and J. Number Theory 113 (2005), 1–9] to investigate Hong’s conjectures. We show that the conjectures of Hong are true for $n\le 3$ but they are both not true for $n\ge 4$.
LA - eng
KW - GCD-closed set; LCM-closed set; greatest-type divisor; divisibility; GCD-closed set; LCM-closed set; greatest-type divisor; divisibility
UR - http://eudml.org/doc/31117
ER -

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