Nonsingularity of least common multiple matrices on gcd-closed sets

Let n be a positive integer. Let S={x1,…,xn} be a set of n distinct positive integers. The least common multiple (LCM) matrix on S, denoted by [S], is defined to be the n×n matrix whose (i,j)-entry is the least common multiple [xi,xj] of xi and xj. The set S is said to be gcd-closed if for any xi,xj∈S,(xi,xj)∈S. For an integer m>1, let ω(m) denote the number of distinct prime factors of m. Define ω(1)=0. In 1997, Qi Sun conjectured that if S is a gcd-closed set satisfying maxx∈S{ω(x)}⩽2, then the LCM matrix [S] is nonsingular. In this paper, we settle completely Sun's conjecture. We show the following result: (i). If S is a gcd-closed set satisfying maxx∈S{ω(x)}⩽2, then the LCM matrix [S] is nonsingular. Namely, Sun's conjecture is true; (ii). For each integer r⩾3, there exists a gcd-closed set S satisfying maxx∈S{ω(x)}=r, such that the LCM matrix [S] is singular.