摘要
Abstract
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.关键词
gcd闭集/极大型因子/最小公倍数矩阵/幂LCM矩阵Key words
gcd-closed set/greatest-type divisor/least common multiple matrix/power LCM matrix分类
数理科学
