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

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

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|d,则det(S^a)|det(S^a),det[S^a]|det[S^b]和det(S^b)|det[S^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.

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),det[S^a]]det[S^b],det(S^a)]det[S^b].But such factorizations fail to be true if S consists of two divisor chains which are not coprime.