Divisibility among determinants of power matrices on gcd-closed sets

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.

Key words:Divisibility, Power GCD matrix, Power LCM matrix, GCD-closed set, r-fold gcd-closed set