On the factorization of LCM matrices on gcd-closed sets. (English) Zbl 0995.15006

Let \(S=\{x_1,\dots, x_n\}\) be a set of \(n\) distinct positive integers. The matrix having the greatest common divisor (GCD) \((x_i,x_j)\), respectively the least common multiple (LCM) \([x_i,x_j]\), of \(x_i\) and \(x_j\) as its \(i,j\)-entry is called the greatest common divisor matrix, denoted by \((S)_n\), respectively the least common multiple matrix, denoted by \([S]_n\). The set is said to be gcd-closed if \((x_i,x_j)\in S\) for all \(1\leq i,j\leq n\).
The author shows that if \(n\leq 3\), then for any gcd-closed set \(S=\{x_1, \dots,x_n\}\), the GCD matrix on \(S\) divides the LCM matrix on \(S\) in the ring \(M_n(Z)\) of \(n\times n\) matrices over the integers. For \(n\geq 4\), there exists a gcd-closed set \(S=\{x_1,\dots,x_n\}\) such that the GCD matrix on \(S\) does not divide the LCM matrix on \(S\) in the ring \(M_n(Z)\). This solves a conjecture raised by the author in his Ph.D. thesis [Some problems related to matrices in number theory. Sichuan Univ. (1998)].


15A23 Factorization of matrices
11C20 Matrices, determinants in number theory
11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors
15B36 Matrices of integers
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[1] Apostol, T.M., Introduction to analytic number theory, (1976), Springer New York · Zbl 0335.10001
[2] Beslin, S., Reciprocal GCD matrices and LCM matrices, Fibonacci quart., 29, 271-274, (1991) · Zbl 0738.11026
[3] Beslin, S.; Ligh, S., Greatest common divisor matrices, Linear algebra appl., 118, 69-76, (1989) · Zbl 0672.15005
[4] Beslin, S.; Ligh, S., Another generalization of Smith’s determinant, Bull. austral. math. soc., 40, 3, 413-415, (1989) · Zbl 0675.10002
[5] Beslin, S.; Ligh, S., GCD-closed sets and the determinants of GCD matrices, Fibonacci quart., 30, 157-160, (1992) · Zbl 0752.11012
[6] Bourque, K.; Ligh, S., On GCD and LCM matrices, Linear algebra appl., 174, 65-74, (1992) · Zbl 0761.15013
[7] Bourque, K.; Ligh, S., Matrices associated with classes of arithmetical functions, J. number theory, 45, 367-376, (1993) · Zbl 0784.11002
[8] Haukkanen, P., Higher dimensional GCD matrices, Linear algebra appl., 170, 53-63, (1992) · Zbl 0749.15013
[9] Hong, S., LCM matrix on an r-fold gcd-closed set, J. sichuan univ. natu. sci. ed., 33, 6, 650-657, (1996) · Zbl 0869.11021
[10] Hong, S., On LCM matrices on GCD-closed sets, Southeast Asian bull. math., 22, 381-384, (1998) · Zbl 0936.15011
[11] S. Hong, Some problems related to matrices in number theory, Ph.D. thesis, Sichuan University, 1998
[12] Hong, S., On the bourque – ligh conjecture of least common multiple matrices, J. algebra, 218, 216-228, (1999), A sketch was given in Advances in Mathematics (China), 25(6) (1996) 566-568 · Zbl 1015.11007
[13] Li, Z., The determinants of GCD matrices, Linear algebra appl., 134, 137-143, (1990) · Zbl 0703.15012
[14] Smith, H.J.S., On the value of a certain arithmetical determinant, Proc. London math. soc., 7, 208-212, (1875-1876)
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