New results on nonsingular power LCM matrices

Let e and n be positive integers and S = {x1, . . . , xn} be a set of n distinct positive integers. The n × n matrix having eth power [xi, xj ] of the least common multiple of xi and xj as its (i, j)-entry is called the eth power least common multiple (LCM) matrix on S, denoted by ([S]). The set S is said to be gcd closed (respectively, lcm closed) if (xi, xj) ∈ S (respectively, [xi, xj ] ∈ S) for all 1 ≤ i, j ≤ n. In 2004, Shaofang Hong showed that the power LCM matrix ([S]) is nonsingular if S is a gcd-closed set such that each element of S holds no more than two distinct two prime factors. In this paper, this result is improved by showing that if S is a gcd-closed set such that every element of S contains at most two distinct prime factors or is of the form pqr with p, q and r being distinct primes and 1 ≤ l ≤ 4 being an integer, then except for the case that e = 1 and 270, 520, 810, 1040 ∈ S, the power LCM matrix ([S]) on S is nonsingular. This gives an evidence to a conjecture of Hong raised in 2002. For the lcm-closed case, similar results are established.