| Title:
|
On Hong’s conjecture for power LCM matrices (English) |
| Author:
|
Cao, Wei |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
57 |
| Issue:
|
1 |
| Year:
|
2007 |
| Pages:
|
253-268 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A set $\mathcal{S}=\lbrace x_1,\ldots ,x_n\rbrace $ of $n$ distinct positive integers is said to be gcd-closed if $(x_{i},x_{j})\in \mathcal{S}$ for all $1\le i,j\le n $. Shaofang Hong conjectured in 2002 that for a given positive integer $t$ there is a positive integer $k(t)$ depending only on $t$, such that if $n\le k(t)$, then the power LCM matrix $([x_i,x_j]^t)$ defined on any gcd-closed set $\mathcal{S}=\lbrace x_1,\ldots ,x_n\rbrace $ is nonsingular, but for $n\ge k(t)+1$, there exists a gcd-closed set $\mathcal{S}=\lbrace x_1,\ldots ,x_n\rbrace $ such that the power LCM matrix $([x_i,x_j]^t)$ on $\mathcal{S}$ is singular. In 1996, Hong proved $k(1)=7$ and noted $k(t)\ge 7$ for all $t\ge 2$. This paper develops Hong’s method and provides a new idea to calculate the determinant of the LCM matrix on a gcd-closed set and proves that $k(t)\ge 8$ for all $t\ge 2$. We further prove that $k(t)\ge 9$ iff a special Diophantine equation, which we call the LCM equation, has no $t$-th power solution and conjecture that $k(t)=8$ for all $t\ge 2$, namely, the LCM equation has $t$-th power solution for all $t\ge 2$. (English) |
| Keyword:
|
gcd-closed set |
| Keyword:
|
greatest-type divisor(GTD) |
| Keyword:
|
maximal gcd-fixed set(MGFS) |
| Keyword:
|
least common multiple matrix |
| Keyword:
|
power LCM matrix |
| Keyword:
|
nonsingularity |
| MSC:
|
11A25 |
| MSC:
|
11C20 |
| idZBL:
|
Zbl 1174.11030 |
| idMR:
|
MR2309964 |
| . |
| Date available:
|
2009-09-24T11:45:33Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128170 |
| . |
| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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