New results on the divisibility of power GCD and power LCM matrices
Let $ a, b $ and $ n $ be positive integers and let $ S $ be a set consisting of $ n $ distinct positive integers $ x_1, ..., x_{n-1} $ and $ x_n $. Let $ (S^a) $ (resp. $ [S^a] $) denote the $ n\times n $ matrix having $ \gcd(x_i, x_j)^a $ (resp. $ {\rm lcm}(x_i, x_j)^a $) as its $ (i, j) $-entry....
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| Language: | English |
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AIMS Press
2022-08-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20221003?viewType=HTML |
| _version_ | 1811286979906633728 |
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| author | Guangyan Zhu Mao Li Xiaofan Xu |
| author_facet | Guangyan Zhu Mao Li Xiaofan Xu |
| author_sort | Guangyan Zhu |
| collection | DOAJ |
| description | Let $ a, b $ and $ n $ be positive integers and let $ S $ be a set consisting of $ n $ distinct positive integers $ x_1, ..., x_{n-1} $ and $ x_n $. Let $ (S^a) $ (resp. $ [S^a] $) denote the $ n\times n $ matrix having $ \gcd(x_i, x_j)^a $ (resp. $ {\rm lcm}(x_i, x_j)^a $) as its $ (i, j) $-entry. For any integer $ x\in S $, if $ (y < x, y|z|x \ {\rm and} \ y, z\in S)\Rightarrow z\in\{y, x\} $, then $ y $ is called a greatest-type divisor of $ x $ in $ S $. In this paper, we establish some results about the divisibility between $ (S^a) $ and $ (S^b) $, between $ (S^a) $ and $ [S^b] $ and between $ [S^a] $ and $ [S^b] $ when $ a|b $, $ S $ is gcd closed (i.e., $ \gcd(x_i, x_j)\in S $ for all $ 1\le i, j\le n $), and $ \max_{x\in S}\{|\{y\in S: y \ \text{is a greatest-type divisor of} \ x \ {\rm in} \ S\}|\} = 2 $. |
| first_indexed | 2024-04-13T03:09:30Z |
| format | Article |
| id | doaj.art-1e88b57865e549d794699f809431034b |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-13T03:09:30Z |
| publishDate | 2022-08-01 |
| publisher | AIMS Press |
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| series | AIMS Mathematics |
| spelling | doaj.art-1e88b57865e549d794699f809431034b2022-12-22T03:05:06ZengAIMS PressAIMS Mathematics2473-69882022-08-01710182391825210.3934/math.20221003New results on the divisibility of power GCD and power LCM matricesGuangyan Zhu 0Mao Li1Xiaofan Xu21. School of Teacher Education, Hubei Minzu University, Enshi 445000, China2. School of Mathematics and Statistics, Southwest University, Chongqing 400715, China3. School of Big Data and Statistics, Sichuan Tourism University, Chengdu 610100, ChinaLet $ a, b $ and $ n $ be positive integers and let $ S $ be a set consisting of $ n $ distinct positive integers $ x_1, ..., x_{n-1} $ and $ x_n $. Let $ (S^a) $ (resp. $ [S^a] $) denote the $ n\times n $ matrix having $ \gcd(x_i, x_j)^a $ (resp. $ {\rm lcm}(x_i, x_j)^a $) as its $ (i, j) $-entry. For any integer $ x\in S $, if $ (y < x, y|z|x \ {\rm and} \ y, z\in S)\Rightarrow z\in\{y, x\} $, then $ y $ is called a greatest-type divisor of $ x $ in $ S $. In this paper, we establish some results about the divisibility between $ (S^a) $ and $ (S^b) $, between $ (S^a) $ and $ [S^b] $ and between $ [S^a] $ and $ [S^b] $ when $ a|b $, $ S $ is gcd closed (i.e., $ \gcd(x_i, x_j)\in S $ for all $ 1\le i, j\le n $), and $ \max_{x\in S}\{|\{y\in S: y \ \text{is a greatest-type divisor of} \ x \ {\rm in} \ S\}|\} = 2 $.https://www.aimspress.com/article/doi/10.3934/math.20221003?viewType=HTMLdivisibilitypower matrixgreatest-type divisorgcd-closed set |
| spellingShingle | Guangyan Zhu Mao Li Xiaofan Xu New results on the divisibility of power GCD and power LCM matrices AIMS Mathematics divisibility power matrix greatest-type divisor gcd-closed set |
| title | New results on the divisibility of power GCD and power LCM matrices |
| title_full | New results on the divisibility of power GCD and power LCM matrices |
| title_fullStr | New results on the divisibility of power GCD and power LCM matrices |
| title_full_unstemmed | New results on the divisibility of power GCD and power LCM matrices |
| title_short | New results on the divisibility of power GCD and power LCM matrices |
| title_sort | new results on the divisibility of power gcd and power lcm matrices |
| topic | divisibility power matrix greatest-type divisor gcd-closed set |
| url | https://www.aimspress.com/article/doi/10.3934/math.20221003?viewType=HTML |
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