On the divisibility of power LCM matrices by power GCD matrices

Let S = {x1, ..., xn} be a set of n distinct positive integers and e ⩾ 1 an integer. Denote the n × n power GCD (resp. power LCM) matrix on S having the e-th power of the greatest common divisor (xi, xj) (resp. the e-th power of the least common multiple [xi, xj]) as the (i, j)-entry of the matrix by ((xi, xj)e) (resp. ([xi, xj]e)). We call the set S an odd gcd closed (resp. odd lcm closed) set if every element in S is an odd number and (xi, xj) ∈ S (resp. [xi, xj] ∈ S) for all 1 ⩽ i, j ⩽ n. In studying the divisibility of the power LCM and power GCD matrices, Hong conjectured in 2004 that for any integer e ⩾ 1, the n × n power GCD matrix ((xi, xj)e) defined on an odd-gcd-closed (resp. odd-lcm-closed) set S divides the n × n power LCM matrix ([xi, xj]e) defined on S in the ring Mn(ℤ) of n × n matrices over integers. In this paper, we use Hong’s method developed in his previous papers [J. Algebra 218 (1999) 216–228; 281 (2004) 1–14, Acta Arith. 111 (2004), 165–177 and J. Number Theory 113 (2005), 1–9] to investigate Hong’s conjectures. We show that the conjectures of Hong are true for n ⩽ 3 but they are both not true for n ⩾ 4.