幂GCD矩阵与幂LCM矩阵的行列式的整除性

设S={x_1,x_2,…,x_n)是由n个不同的正整数组成的集合,并设整数a≥1,如果n阶矩阵的第i行j列元素是S中元素x_i和x_j的最大公因子的a次幂(x_i,x_j)~a,则称该矩阵是定义在S上的口次幂GCD矩阵,用(S~a)表示.类似定义幂LCM矩阵S~a].本文证明了:设S是由n个不同的正整数组成的一个最大公因子封闭集,且正整数a∣b.如果n≤3,那么det(S~a)I detS~b];如果max{x_i)＜12,那么det(S~a)f detS~b].x_i∈S

Divisibility properties of determinants of power GCD and determinants of power LCM matrices

Let S= {x_1 ,x_2, …,x_n} be a set of n distinct positive integers and a≥1 an integer. The matrix having the a-th power (x_i ,x_j)~a of the greatest common divisor of x_i and x_j as its i,j-entry is called a-th power greatest common divisor (GCD) matrix defined on S~a denoted by (S~a). Similarly we can define the a-th power LCM matrixS~a]. In this paper, we prove that if S is a GCD-closed set with | S | ≤3 and a ∣ b, then det(S~a) ∣det S~b], We also show that if S is a gcd-closed set satisfying max(S)＜12 and a∣b, then det (S~a)∣detS~b].