On Nonsingular Power LCM Matrices

Let e ≥ 1 be an integer and S={x 1 ,…,x n } a set of n distinct positive integers. The matrix ([x i , x j ] e ) having the power [x i , x j ] e of the least common multiple of x i and x j as its (i, j)-entry is called the power least common multiple (LCM) matrix defined on S. The set S is called gcd-closed if (x i ,x j ) ∈ S for 1≤ i, j≤ n. Hong in 2004 showed that if the set S is gcd-closed such that every element of S has at most two distinct prime factors, then the power LCM matrix on S is nonsingular. In this paper, we use Hong's method developed in his previous papers to consider the next case. We prove that if every element of an arbitrary gcd-closed set S is of the form pqr, or p 2 qr, or p 3 qr, where p, q and r are distinct primes, then except for the case e=1 and 270, 520 ∈ S, the power LCM matrix on S is nonsingular. We also show that if S is a gcd-closed set satisfying x i < 180 for all 1≤ i≤ n, then the power LCM matrix on S is nonsingular. This proves that 180 is the least primitive singular number. For the lcm-closed case, we establish similar results.