## 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.

### MSC:

 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

### Citations:

Zbl 0761.15013; Zbl 0869.11021; Zbl 0869.11022