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幂GCD矩阵及幂LCM矩阵的行列式的非整除性 被引量：2

Non-Divisibility of the Determinants of Power GCD and Power LCM Matrices
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摘要 当S是任一因子链,且|S|≥2时,给出了幂GCD矩阵及幂LCM矩阵的行列式的计算公式,并且得到了一个关于其行列式的非整除性的结果. The authors give the computational formula and obtain a result on non-divisibility of determinants of power GCD matrices and power LCM matrices when it is a divisor chain of length not less than 2.
作者 谭千蓉 林宗兵
机构地区 攀枝花学院计算机学院
出处 《武汉大学学报（理学版）》 CAS CSCD 北大核心 2010年第4期383-386,共4页 Journal of Wuhan University:Natural Science Edition
基金 教育部新世纪优秀人才支持计划基金资助项目(NCET-06-0785)
关键词 整除 因子链 幂GCD矩阵 幂LCM矩阵 divisibility divisor chain power GCD matrix power LCM matrix
分类号 O156 [理学—基础数学]
  • 相关文献

参考文献22

  • 1Smith H J. On the value of a certain arithmetical determinant[J]. Proc London Math Soc,1875-1876,7: 208-212.
  • 2Bourque K, Ligh S. On GCD and LCM matrices[J]. Linear Algebra Appl , 1992,174 : 65-74.
  • 3Bourque K,Ligh S. Matrices associated with arithmetical functions[J]. Linear Algebra,1993,34(3):261- 267.
  • 4Cao W. On Hong's conjecture for power LCM matrices[J]. Czechoslovak Math ,2007,57:253-268.
  • 5Codeca P,Nair M. Calculating a determinant associated with multiplicative functions [J]. Boll Unione Mat ltal Sez B Artic Ric Mat ,2002,5:545-555.
  • 6Haukkanen P, Korkee I. Notes on the divisibility of LCM and GCD matrices[J]. International J Math and Math Science, 2005,6:925-935.
  • 7何聪.最大公因数闭集上幂矩阵的行列式整除性[J].数学学报（中文版）,2006,49(3):647-650. 被引量：1
  • 8He C,Zhao J. More on divisibility of determinants of LCM matrices on GCD-closed sets [J]. Southeast Asian Bull Math ,2005,29 :887-893.
  • 9Hilberdink T. Determinants of multiplicative toeplitz matrices[J]. Acta Arith, 2006,125(3):265-284.
  • 10Hong S F. On the bourque-ligh conjecture of least common multiple matrices [J]. Agebra, 1999, 218: 216- 228.

二级参考文献19

  • 1Smith H.J.S.,On the value of a certain arithmetical determinant,Proc.London Math.Soc.,1875/1876,7:208-212.
  • 2Beslin S.and Ligh S.,Another generalization of Smith's determinant,Bull.Austral.Math.Soc.,1989,40:413-415.
  • 3Bourque K.and Ligh S.,On GCD and LCM matrices,Linear Algebra Appl.,1992,174:65-74.
  • 4Hong S.,Divisibility of determinants of least common multiple matrices on GCD-closed Sets,Southeast Asian Bulletin of Mathematics,2003,27:615-621.
  • 5Apostol T.M.,Arithmetical properties of generalized Ramanujan sums,Pacific J.Math.,1972,41:281-293.
  • 6Beslin S.and Ligh S.,Greateat common matrices,Linear Algebra Appl.,1989,118:69-76.
  • 7Bourque K.and Ligh S.,Matrices associated with classes of arithmetical function,J.Number Theory,1993,45:367-376.
  • 8Hong S.,LCM matrix on an r-fold gcd-closed set,J.Sichuan Univ.Natu.Sci.Ed.,1996,33:650-657.
  • 9Hong S.,Bounds for determinants of matrices associated with classes of arithmetical functions,Linear Algebra Appl.,1998,281:311-322.
  • 10Hong S.,On the Bourque-Ligh conjecture of Least common multiple matrices,J.Algebra,1999,218:216-228;A sketch of proofs was given in Adv.in Math.China,1996,25:566-568.

同被引文献38

  • 1谭千蓉,林宗兵,刘浏.两个互素因子链上的幂GCD矩阵的行列式与幂LCM矩阵的行列式的整除性[J].四川大学学报（自然科学版）,2009,46(6):1581-1584. 被引量：6
  • 2Beslin S, Ligh S. Another generalisation of Smith's determinant [J]. Bull Austral Math Soc, 1989, 40: 413.
  • 3Bourque K, Ligh S. On GCD and LCM matrices[J]. Linear Algebra Appl, 1992, 174: 65.
  • 4Cao W. On HonKs conjecture for power LCM matrices [J]. Czechoslovak Math, 2007, 57: 253.
  • 5Feng W, Hong S, Zhao J. Divisibility properties of power LCM matrices by power GCD matrices on gedelosed sets [J]. Discrete Math, 2009, 309: 2627.
  • 6He C. Divisibility of determinants of power matriees on GCD-elosed sets [J]. Aeta Math, 2006, 49: 647.
  • 7He C, Zhao J. More on divisibility of determinants of LCM matrices on GCD-elosed sets[J]. Southeast Asian Bull Math, 2005, 29.. 887.
  • 8Hong S. Ged-elosed sets and determinants of matrices associated with arithmetical functions [J]. Acta Arith, 2002, 101: 321.
  • 9Hong S. On the factorization of LCM matrices on ged-closed sets [J]. Linear Algebra Appl, 2002, 345 : 225.
  • 10Hong S. Factorization of matrices associated with classes of arithmetical functions [J]. Colloq. Math, 2003, 98:113.

引证文献2

二级引证文献3

武汉大学学报（理学版）
2010年 第4期

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