对洪关于幂LCM矩阵的一个猜想的注记

一个含有n个不同正整数的集合S={xt，…，xn}称为是gcd闭的，如果S中任两个整数的最大公因子也在S中，洪绍方在2002年猜想：对于给定的一个正整数t，存在一个仅由t决定的正整数k(t)，使得当n≤k(t)时，定义在任意gcd闲集S={xt，…，xn}上的幂LCM矩阵(xi，xj]^t)是非奇异的；而当n≥k(t) 1，则存在一个gcd闭集S={xt，…，xn}，使得定义在其上的幂LCM矩阵(xi，xj]^t)奇异，洪于1999年证明了k (1)=7，在本文中，作者证明了若t≥2，则有k(t)≥8．

Remarks on a Conjecture of Hong of Power LCM Matrices

A set S={x1,...,xn} of n distinct positive integers is said to be gcd-closed if (xi, xj)∈S for all 1≤i, j≤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≤k(t), then the power LCM matrix (xi, xj]t) defined on any gcd-closed set S={x1,...,xn} is nonsingular; but for n≥k(t)+1, there exists a gcd-closed set S={x1,...,xn} such that the power LCM matrix (xi, xj]t) on S is singular. In 1999, Hong proved that k(1)=7. In this paper, the author showed that k(t)≥8 for any positive integer t≥2.