Bourque and Ligh (1992) conjectured that the least common multiple matrix on a GCD-closed set 
is nonsingular. This conjecture was shown to be false by Haukkanen et al. (1997)
and subsequently by Hong (1999).
Bourque-Ligh Conjecture
See also
GCD-Closed Set, Least Common Multiple MatrixExplore with Wolfram|Alpha
References
Bourque, K. and Ligh, S. "On GCD and LCM Matrices." Linear Algebra Appl. 174, 65-74, 1992.Haukkanen, P.; Wang, J.; and Sillanpää, J. "On Smith's Determinant." Linear Alg. Appl. 258, 251-269, 1997.Hong, S. "On the Bourque-Ligh Conjecture of Least Common Multiple Matrices." J. Algebra 218, 216-228, 1999.Referenced on Wolfram|Alpha
Bourque-Ligh ConjectureCite this as:
Weisstein, Eric W. "Bourque-Ligh Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Bourque-LighConjecture.html