GCD matrix

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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.

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).