LCM matrix on an \(r\)-fold GCD-closed set. (English) Zbl 0869.11021

Let \(S=\{x_1,x_2, \dots, x_n\}\) be a set of distinct positive integers. The LCM matrix on \(S\) is the \(m\times n\) matrix whose \(i\), \(j\)-entry is the least common multiple of \(x_i\) and \(x_j\). K. Bourque and S. Ligh [Linear Algebra Appl. 174, 65-74 (1992; Zbl 0761.15013)] conjectured that the LCM matrix on a GCD-closed set is nonsingular. P. Haukkanen, J. Wang and J. Sillanpää [Linear Algebra Appl. 258, 251–269 (1997)] show that this conjecture does not hold by giving a counterexample with \(n=9\).
The present author introduces the concept of an \(r\)-fold GCD-closed set in order to investigate the conjecture. The author shows that if \(n\leq 5\), then the conjecture holds. If \(n\geq 6\) and \(S\) is an \((n-5)\)-fold GCD-closed set, then the LCM matrix on \(S\) is nonsingular. In a later paper (see the following review Zbl 0869.11022) the author solves completely a problem relating to the Bourque-Ligh conjecture.


11C20 Matrices, determinants in number theory
11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors
11A25 Arithmetic functions; related numbers; inversion formulas
15B36 Matrices of integers