On Power LCM Matrices And LCM Equation On Gcd-closed Sets

Let S = {x₁,...x_n} be a set of n distinct positive integers. The set S is said tobe gcd-closed if (x_i,x_j) ∈S for all 1 ≤i,j ≤n. The matrix having the e-powerleast common multiple [x_i,x_j]^e of xi and xj as its i,j-entry is called the e-powerleast common multiple (LCM) matrix, denoted by ([x_i,x_j]^e). In this paper, we showthat for any real number e ≥1 and n ≤7, the power LCM matrix ([x_i,x_j]^e) definedon any gcd-closed set S = {x₁,...,x_n} is nonsingular. This confirms a conjectureraised by Hong in 2004 if n ≤7 and e ≥1.Hong conjectured in 2002 that for a given positive integer t there is a positiveinteger k(t) depending only on t, such that if n ≤k(t), then the power LCM matrix([x_i,x_j]^t) defined on any gcd-closed set S = {x₁,...,x_n} is nonsingular, but forn ≥k(t) + 1, there exists a gcd-closed set S = {x1,...,xn} such that the powerLCM matrix ([x_i,x_j]^t) on S is singular. In 2004, Cao proved k(t) ≥9 if and onlyif the following Diophantine equation (LCM equation) has no t?th power solutionsunder the certain constraints:Let ω(x) denote the number of distinct prime divisors of a given integer x andy = [y₁,y₂,y₃,y₄]. This paper prove that if ω(y) < 4, (1') has no t(≥2)?th powersolutions and give the necessities to its 2-th power solutions when ω(y) = 4. Wefurther prove that if y ≤1334025, (1') has no 2-th power solutions and henceconjecture that k(2) ≥9.