摘要
设S={x1,…,xn}是由n个不同正整数组成的最大公因数闭集.得到的主要结果是:(1)如果n≤3,则det(S)n2整除det[S]n2;(2)如果max{xi}xi∈S<12,则det(S)2n整除det[S]2n;(3)当n=4时,存在最大公因数闭集S,有det(S)2n不整除det[S]n2.
Let S = {x1,…,xn} be a GCD-closed set of n distinct positive integers. In this paper, our main results are as follows: ( 1 ) If n ≤3 then the determinant det(S)n^2 of the GED quadratic matrix (S)n^2 on S divides the determinant det[ S] n^2 of the LCM quadratic matrix [ S ]n^2 on S; ( 2 ) If max { xi xi∈S } 〈 12, then the determinant det (S) n^2 of the GED quadratic matrix (S) n^2 on S divides the determinant det [S]n^2 of the LCM quadratic matrix [ S]n^2 on S; (3) If n =4, there exists a GCD-elosed set S = {x1 …,xn} , such that the determinant det(S)n^2 of the GCD quadratic matrix (S)n^2 on S does not divide the determinant det[S]n^2 of the LCM quadratic matrix [S]n^2 on S.
出处
《四川师范大学学报(自然科学版)》
CAS
CSCD
北大核心
2006年第3期300-302,共3页
Journal of Sichuan Normal University(Natural Science)
基金
四川省教育厅自然科学重点基金资助项目
关键词
最大公因数闭集
最大公因数平方矩阵
最小公倍数平方矩阵
行列式
整除性
GCD-closed set
Greatest common divisor quadratic matrix
Least common multiple quadratic matrix
Determinant
Divisibility