Notes on Hong's conjecture on nonsingularity of power LCM matrices

Guangyan Zhu; Kaimin Cheng; Wei Zhao; Guangyan Zhu; Kaimin Cheng; Wei Zhao

doi:10.3934/math.2022572

AIMS Mathematics

2022, Volume 7, Issue 6: 10276-10285. doi: 10.3934/math.2022572

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Research article

Notes on Hong's conjecture on nonsingularity of power LCM matrices

1.
School of Teacher Education, Hubei Minzu University, Enshi 445000, China
2.
School of Mathematics and Information, China West Normal University, Nanchong 637009, China
3.
Science and Technology on Communication Security Laboratory, Chengdu 610041, China

Received: 17 January 2022 Revised: 08 March 2022 Accepted: 10 March 2022 Published: 23 March 2022
MSC : Primary 11C20; Secondary 11A05, 15B36

Let $ a, n $ be positive integers and $ S = \{x_1, ..., x_n\} $ be a set of $ n $ distinct positive integers. The set $ S $ is said to be gcd (resp. lcm) closed if $ \gcd(x_i, x_j)\in S $ (resp. $ [x_i, x_j]\in S $) for all integers $ i, j $ with $ 1\le i, j\le n $. We denote by $ (S^a) $ (resp. $ [S^a] $) the $ n\times n $ matrix having the $ a $th power of the greatest common divisor (resp. the least common multiple) of $ x_i $ and $ x_j $ as its $ (i, j) $-entry. In this paper, we mainly show that for any positive integer $ a $ with $ a\ge 2 $, the power LCM matrix $ [S^a] $ defined on a certain class of gcd-closed (resp. lcm-closed) sets $ S $ is nonsingular. This provides evidences to a conjecture raised by Shaofang Hong in 2002.
- power LCM matrix,
- power GCD matrix,
- nonsingularity,
- gcd-closed set
Citation: Guangyan Zhu, Kaimin Cheng, Wei Zhao. Notes on Hong's conjecture on nonsingularity of power LCM matrices[J]. AIMS Mathematics, 2022, 7(6): 10276-10285. doi: 10.3934/math.2022572

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Abstract

Let $ a, n $ be positive integers and $ S = \{x_1, ..., x_n\} $ be a set of $ n $ distinct positive integers. The set $ S $ is said to be gcd (resp. lcm) closed if $ \gcd(x_i, x_j)\in S $ (resp. $ [x_i, x_j]\in S $) for all integers $ i, j $ with $ 1\le i, j\le n $. We denote by $ (S^a) $ (resp. $ [S^a] $) the $ n\times n $ matrix having the $ a $th power of the greatest common divisor (resp. the least common multiple) of $ x_i $ and $ x_j $ as its $ (i, j) $-entry. In this paper, we mainly show that for any positive integer $ a $ with $ a\ge 2 $, the power LCM matrix $ [S^a] $ defined on a certain class of gcd-closed (resp. lcm-closed) sets $ S $ is nonsingular. This provides evidences to a conjecture raised by Shaofang Hong in 2002.

References

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