Nonsingularity of least common multiple matrices on gcd-closed sets

Abstract

Let n be a positive integer. Let S = { x 1 , … , x n } 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 [ x i , x j ] of x i and x j . The set S is said to be gcd-closed if for any x i , x j ∈ S , ( x i , x j ) ∈ 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 max x ∈ 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 max x ∈ 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 max x ∈ S { ω ( x ) } = r , such that the LCM matrix [ S ] is singular.