Nonsingularity of least common multiple matrices on gcd-closed sets

✍ Scribed by Shaofang Hong

• Publisher: Elsevier Science
• Year: 2005
• Tongue: English
• Weight: 183 KB
• Volume: 113
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2005.03.004

✦ Synopsis

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 (m) denote the number of distinct prime factors of m. Define (1) = 0. In 1997, Qi Sun conjectured that if S is a gcd-closed set satisfying max x∈S { (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 { (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 { (x)} = r, such that the LCM matrix [S] is singular