关于平方LCM矩阵和LCM方程的注记  

Notes on Square LCM Matries and LCM Equations

作　　者：李懋[1] 曹炜[1] 

机构地区：[1]四川大学数学学院,成都610064

出　　处：《四川大学学报（自然科学版）》2005年第2期240-244,共5页Journal of Sichuan University(Natural Science Edition)

摘　　要：设S={x1,…,xn}是由n个不同正整数组成的集合.第i行j列元素为xi和xj的最小公倍数[xi,xj]的n×n阶矩阵([xi,xj])称为定义在S上的LCM矩阵.如果对所有的1≤i,j≤n,有(xi,xj)∈S,称S是最大公因子封闭的(gcd closed).作者考虑了方程11+1(y2,y3)=0[y1,y2,y3,y4]-∑4(y1,y3)+1(y1,y2)+1yii=1的二次幂整数解,证明了对于给定的整数x,如果用ω(x)表示x的不同素因子的个数并令y=[y1,y2,y3,y4],那么当ω(y)<4时,方程没有t(≥2)次幂整数解,并且给出ω(y)=4时方程有二次幂整数解的必要条件.进一步证明了y≤1334025时方程无二次幂整数解.Let S={x1,…,xn} be a set of n distinct positive integers. The matrix ([xi, xj]) having the least common multiple [xi, xj] of xi and xj as its i,j-entry is called the least common multiple (LCM) matrix on S. If (xi, xj)∈S for all 1≤ i,j≤ n, then S is said to be gcd-closed. Hong conjectured in 2002 that for a given positive integer t there is a positive integer k(t) depending only on t, such that if n≤ k(t), then the power LCM matrix (([xi, xj]t)) defined on any gcd-closed set S={x1,…,xn} is nonsingular, but for n≥ k(t) + 1, there exists a gcd,closed set S={x1,…,xn} such that the power LCM matrix (([xi, xj]t)) on S is singular. In 1999, Hong proved that k(1)=7. In 2003, Cao proved that k(t)≥8 for all t≥2, but the proof was too complicated. Recently, Cao provided a relatively brief proof for k(t)≥8, (t≥2) and proved that k(t)≥9 iff the following Diophantine equation (LCM equation) has no t-th power solutions under the certain constraints:1[y1,y2,y3,y4]-∑4i=11yi+1(y1,y2)+1(y1,y3)+1(y2,y3)=0.The authors mainly discuss the 2-th power solutions of this equation. Let ω(x) denote the number of distinct prime divisors of a given integer x and y=[y1, y2, y3, y4], they prove that if ω(y)<4, the equation has no t(≥2)-th power solutions and give the necessities to its 2-th power solutions when ω(y)=4. And prove that if y≤1334025, the equation has no 2-th power solutions and hence conjecture that k(2)≥9.

关 键 词：gcd-closed集 (幂)LCM矩阵 LCM方程 

分 类 号：O156.1[理学—数学]