LCM矩阵的一些性质(英文)

SOME PROPERTIES OF LCM MATRICES

设S=｛X1，X2，…，Xn｝是不同正整数的集合.称S为gcd封闭集，如果Xi与xj的最大公因数（Xi，xj）也属于S（1≤i，j≤n）．矩阵［S］被称为S上的最小公倍数（LCM）矩阵，如果它的i，j位置元素是Xi与xj的最小公倍数［xi，xj］．BourqueandLigh猜想：一个gcd封闭集上的LCM矩阵是可逆的．作者证明了当n≤7时猜想成立，但当n≥8时猜想不成立．同时也给出一个新的计算det［S］的公式．

Let S={x1, x2, '', xn} be a set of distinct positive integers. The set S is calledged-closed if it contains the greatest common divisor (xi, xj) of xi. and xj for 1<i, j<n. The matrix [S]is called the least common multiple (LCM) matrix on S if its i, j entry is the least common multiPle [xi, xj] of xi and xi. Bourque and Ligh conjecturedthat the LCM matrixon a ged-closed Setis invertible. The aimof this paper is to show that the conjecture does not hold in general when n≥8,but the LCM matrix on a geb-closed set is invertible if n≤7. A new calculating formula of det [S]is also presented.