On the Bourque–Ligh Conjecture of Least Common Multiple Matrices

Let S = {x1,…,xn} be a set of n distinct positive integers. The matrix [S]n having the least common multiple [xi, xj] of xi and xj as its i, j-entry is called the least common multiple (LCM) matrix on S. A set S is gcd-closed if (xi, xj) S for 1 ≤ i, j ≤ n. Bourque and Ligh conjectured that the LCM matrix [S]n, defined on a gcd-closed set S, is nonsingular. In this paper we prove that the conjecture is true for n ≤ 7 and is not true for n ≥ 8. So the conjecture is solved completely.