LCM矩阵的可逆性与不定方程ayz+bzx+cxy=xyz+1的解(英文)  

The Invertibility of LCM Matrices and the Solutions of Indeterminate Equation ayz+bzx+cxy=xyz+1

作　　者：王伯英[1] 侯耀平[2] 

机构地区：[1]北京师范大学数学系 [2]湖南师范大学数学系

出　　处：《数学进展》2002年第1期31-36,共6页Advances in Mathematics（China）

基　　金：The project supported by NSFC(No. 19671013).

摘　　要：设Ｓ＝｛ｘ１，ｘ２，…ｘｎ｝是不同正整数的集合. 已经知道当ｎ≤７时在最大公因数封闭集Ｓ上的．ＬＣＭ矩阵是可逆的；也知道当ｎ≥９时有无限多个包含整数１的最大公因数封闭集它们的ＬＣＭ矩阵是奇异的；这篇文章的主要结果是证明当ｎ＝８且包含整数１时，除了２０个最大公因数封闭集外，其余所有最大公因数封闭集上的ＬＣＭ矩阵都是可逆的，而这归结为解一个不定方程．Let S ={x1,x2…,xn} be a set of distinct positive integers. It was known that the LCM matrix on a gcd-closed set S is invertible if n ≤ 7 and when n ≥ 9 there exist infinitely many gcd-closed sets with integer 1 such that their LCM matrices are singular. The main result of this paper is to show that when n = 8 and x8 = 1 the LCM matrices on almost all gcd-closed sets are invertible except 20 gcd-closed sets, and this reduces the solutions of an indeterminate equation.

关 键 词：最大公因数封闭集 最小公倍数矩阵 不定方程 解 LCM矩阵 可逆性 

分 类 号：O156.7[理学—数学]