关于GCD封闭集上幂矩阵行列式之间的整除性

Divisibility among determinants of power matrices on gcd-closed sets

摘要:

设a,b,n为正整数，S={x_1,…,x_n}是由n个不同正整数x_1,…,x_n构成的集合.以(S^a) ([S^a])表示n×n矩阵，其中第i行第j列元为x_i和x_j的最大公因子(x_i,x_j)(最小公倍数[x_i,x_j])的a次幂.本文研究了GCD封闭集上幂矩阵行列式之间的整除性，加强和推广了Hong在2003年和Chen等在2020年得到的结果.

Abstract:

Let a,b,n be positive integers and let S={x_1,…,x_n } be a set of n distinct positive integers. Denoted by (S^a) (resp. [S^a]) the n×n matrix having the ath power of the greatest common divisor (resp. the least common multiple) of x_i and x_j as its (i,j)-entry, we iIn this paper study the divisibility among the determinants of power matrices on GCD-closed sets and extend Hong’s theorem obtained in 2003 and the theorems of Chen and Hong obtained in 2020.