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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)].

MSC:

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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