有限个互素因子链上幂GCD矩阵与幂LCM矩阵的行列式的整除性

Divisibility properties of determinants of power GCD matrices and power LCM matrices on finitely many pairwise relatively prime divisor chains

设S={x1,x2,...,xn}是由n个不同的正整数组成的集合,并设a为正整数.如果一个n阶矩阵的第i行j列元素是S中元素xi和xj的最大公因子的a次幂(xi,xj)a,则称该矩阵为定义在S上的a次幂最大公因子(GCD)矩阵,用(Sa)表示;类似定义a次幂LCM矩阵[Sa].如果存在{1,2,...,n}上的一个置换σ使得xσ(1)|xσ(2)|···|xσ(n),则称S为一个因子链.如果存在正整数k,使得S=S1∪S2∪···∪Sk,其中每一个Si(1ik)均为一个因子链,并且对所有的1i=jk,Si中的每个元素与Sj中的每个元素互素,则称S由有限个互素因子链构成.本文中,设S由有限个互素的因子链构成,并且1∈S.我们首先给出幂GCD矩阵与幂LCM矩阵的行列式的公式,然后证明:如果a|b,则det(Sa)|det(Sb),det[Sa]|det[Sb],det(Sa)|det[Sb].最后我们指出:如果构成S的有限个因子链不互素,则此结论一般不成立.

Let S = （x1, X2 ,…… xn} be a set of n distinct positive integers and a ≥ 1 be an integer. The matrix having the a-th power （xi,xj）a of the greatest common divisor of xi and xj as its i,j-entry is called the a-th power greatest common divisor （GCD） matrix defined on S, denoted by （sa）. Similarly we can define the a-th power LCM matrix [sa]. The set S is called a divisor chain if there exists a permutation σ on {1, 2,..., n} such that xσ（1）｜xσ（2）｜…｜xσ（n）. We say that the set S consists of finitely many pairwise relatively prime divisor chains if there is a positive integer k such that we can partition S as S = S1 ∪ S2 ∪ …∪ Sk, where Si is a divisor chain for each 1 ≤ i ≤k and each element of Si is coprime to each element of Sj for all 1 ≤ i ≠ j ≤ k. In this paper, we first obtain the formulae of determinants of power GCD matrix （Sa） and power LCM matrix [Sa] and then show that det（Sa）ldet（Sb）, det[Sa] ｜det[Sb] and det（Sa） ]det[Sb] if a｜b and S consists of finitely many pairwise relatively prime divisor chains with 1 ∈ S. But such factorizations fail to be true in general if such divisor chains are not pairwise relatively prime.