关于LCM方程的李-曹猜想的注记  

Notes on the Li-Cao Conjecture on LCM Equations

作　　者：方露艳[1] 

机构地区：[1]安徽师范大学数学系,芜湖241000

出　　处：《四川大学学报（自然科学版）》2008年第3期467-470,共4页Journal of Sichuan University(Natural Science Edition)

基　　金：国家自然科学基金(10071001);安徽省自然科学基金(01046103);安徽省教育厅自然科学基金(2002KJ131)

摘　　要：在研究Hong关于定义在gcd封闭集上的幂LCM矩阵[Se](e为正整数)的非奇异性的一个猜想时,李和曹研究了如下的不定方程(称为LCM方程):.他们首先证明了当ω(y)<4时,方程无解,这里y=lcm[y1,y2,y3,y4],ω(y)表示y的不同素因子的个数;然后他们给出ω(y)=4且y=p21p22p32p42m时,方程有2次幂整数解的必要条件,这里pi为不同素数,m≥1;根据这些必要条件他们接着验证了方程当y≤1 334 025时没有2次幂整数解;最后他们提出猜想:若n≤9,则定义在gcd封闭集S={x1,…,xn}上的平方LCM矩阵[S2]是非奇异的,即LCM方程没有2次幂整数解.本文作者推广了李-曹关于LCM方程有2次幂整数解的研究:首先给出了当ω(y)=4且y=p21m1p22m2p23m3p24m4时,方程有2次幂整数解的必要条件,并给出了当ω(y)≥4时,方程解的表达式(如果存在的话),这里pi为不同素数,mi≥1;然后根据这些必要条件在计算机上验证了方程当y≤260 620 460 100时没有2次幂整数解,进一步支持了李-曹猜想.When studying Hong＇s conjecture for the nonsingularity of power LCM matrices defined on any ged-closed sets, Li and Cao considered the following Diophantine equation （LCM equation）： 1/1cm[y1,y2,y3,y4]-∑^4 i=1/yi＋1/gcd（y1,y2）＋1/gcd（y1,y3）＋1/gcd（y2,y3）=0.They first proved that if ω （y） 〈 4, then the equation has no solutions, where y = 1cm[ y1, y2, y3, y4 ] and ω （y） denotes the number of distinct prime divisors of the integer y. Then, they gave necessary conditions for the equation to have 2-nd power solutions when ω（y） = 4 and y = p^2 1p^2 2p^2 3p^2m4, where Pi are distinct primes and m ∈Z^＋ . According to these conditions, they verified that if y ≤1 334 025, then the equation llas no 2- nd power solutions and hence conjectured that if n ≤ 9, power LCM matrices defined on any gcd-colsed set {x1,…, xn } are nonsingular. In this paper, the author generalizes Li-Cao result on the 2-nd power solutions of the equation. She first gives necessary conditions for the equation to have 2-nd power solutions when ω （y）= 4andy =y=p^2m 1,p^2m^2 2p^2^m4 4,where Pi are distinct primes and mi ∈Z^＋ , i = 1,2,3,4, and gives expres sions of its possible 2-nd power solutions when to（y）≥ 4. Then, using these conditions, she verifies on a computer that if y ≤ 260 620 460 100, the equation has no 2-nd power solutions, further supporting Li-Cao Conjecture.

关 键 词：gcd封闭集 (幂)LCM矩阵 LCM方程 

分 类 号：O156.1[理学—数学]