最大公因数闭集上平方矩阵的行列式的整除性

Divisibility of Determinants of Quadratic Matrices on GCD-closed Sets

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.