On Hong’s conjecture for power LCM matrices

@article{Cao2007,
abstract = {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 $t\ge 2$.},
author = {Cao, Wei},
journal = {Czechoslovak Mathematical Journal},
keywords = {gcd-closed set; greatest-type divisor(GTD); maximal gcd-fixed set(MGFS); least common multiple matrix; power LCM matrix; nonsingularity; gcd-closed set; greatest-type divisor (GTD); maximal gcd-fixed set (MGFS)},
language = {eng},
number = {1},
pages = {253-268},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On Hong’s conjecture for power LCM matrices},
url = {http://eudml.org/doc/31128},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Cao, Wei
TI - On Hong’s conjecture for power LCM matrices
JO - Czechoslovak Mathematical Journal
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 1
SP - 253
EP - 268
AB - 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 $t\ge 2$.
LA - eng
KW - gcd-closed set; greatest-type divisor(GTD); maximal gcd-fixed set(MGFS); least common multiple matrix; power LCM matrix; nonsingularity; gcd-closed set; greatest-type divisor (GTD); maximal gcd-fixed set (MGFS)
UR - http://eudml.org/doc/31128
ER -