Nonsingularity of least common multiple matrices on gcd-closed sets |
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Authors: | Shaofang Hong |
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Institution: | Mathematical College, Sichuan University, Chengdu 610064, P.R. China |
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Abstract: | 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. |
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Keywords: | 11C20 11A25 |
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