GCD matrix

In mathematics, a greatest common divisor matrix (sometimes abbreviated as GCD matrix) is a matrix/

GCD matrix example

Let $S=\{x_{1},x_{2},\ldots ,x_{n}\}$ be a set of positive integers. Then the $n\times n$ matrix $(S)$ having the greatest common divisor $(x_{i},x_{j})$ as its $ij$ entry is referred to as the GCD matrix on $S$. The LCM matrix $[S]$ is defined analogously.

Conjectures

Bourque-Ligh Conjecture: Bourque and Ligh (1992) conjectured that the LCM matrix on a GCD-closed set S is nonsingular. This conjecture was shown to be false by Haukkanen, Wang and Sillanpää (1997) and subsequently by Hong (1999).

References

K. Bourque; S. Ligh (1992). "On GCD and LCM matrices". Linear Algebra and Its Applications. 174: 65–74.

P. Haukkanen; J. Wang; J. Sillanpää (1997). "On Smith's determinant". Linear Algebra and Its Applications. 258: 251–269.

S. Hong (1999). "On the Bourque–Ligh conjecture of least common multiple matrices". Journal of Algebra. 218: 216–228.

H. J. S. Smith (1875). "On the value of a certain arithmetical determinant". Proceedings of the London Mathematical Society. 1: 208–213.