| Title:
|
On the divisibility of power LCM matrices by power GCD matrices (English) |
| Author:
|
Zhao, Jianrong |
| Author:
|
Hong, Shaofang |
| Author:
|
Liao, Qunying |
| Author:
|
Shum, K. P. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
57 |
| Issue:
|
1 |
| Year:
|
2007 |
| Pages:
|
115-125 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $n\ge 4$. (English) |
| Keyword:
|
GCD-closed set |
| Keyword:
|
LCM-closed set |
| Keyword:
|
greatest-type divisor |
| Keyword:
|
divisibility |
| MSC:
|
11A25 |
| MSC:
|
11C20 |
| MSC:
|
15A36 |
| idZBL:
|
Zbl 1174.11031 |
| idMR:
|
MR2309953 |
| . |
| Date available:
|
2009-09-24T11:44:21Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128159 |
| . |
| Reference:
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