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 $\gcd(x_{i},x_{j})$ as its $ij$ entry is referred to as the GCD matrix on $S$. The LCM matrix $[S]$ is defined analogously.

The study of GCD type matrices originates from Smith (1875) who evaluated the determinant of certain GCD and LCM matrices. Smith showed among others that the determinant of the $n\times n$ matrix $(\gcd(i,j))$ is $\phi (1)\phi (2)\cdots \phi (n)$, where $\phi$ is Euler’s totient function.

Conjectures

Bourque-Ligh Conjecture: Bourque & 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 & Sillanpää (1997) and subsequently by Hong (1999). A lattice-theoretic approach is provided by Korkee, Mattila & Haukkanen (2019).