两个互素因子链上的幂GCD矩阵的行列式与幂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a次幂最大公因子(GCD)矩阵,用(S~a)表示. 类似可定义a次幂LCM矩阵S~a].作者证明了:设S由两个互素的因子链构成并且1∈S. 若a|b,则det(S~a)|det(S~b),detS~a]|detS~b]和det(S~a)|detS~b].若S由两个不互素的因子链构成, 则如此分解定理不成立.

Divisibility properties of determinants of power GCD matrices and power LCM matrices on two coprime divisor chains

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, denoted by (S~a).Similarly we can define the a-th power LCM matrix S~a].In this paper, the authors prove that if S consists of two coprime divisor chains and 1∈S, a|b, then det(S~a)|det(S~b),detS~a]|detS~b],det(S~a)|detS~b].But such factorizations fail to be true if S consists of two divisor chains which are not coprime.