New results on nonsingular power LCM matrices

Abstract

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) e 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) e ). The set S is said to be gcd closed (respectively, lcm closed) if (xi,xj) 2 S (respectively, (xi,xj) 2 S) for all 1 � i, jn. In 2004, Shaofang Hong showed that the power LCM matrix ((S) e ) 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 p l qr 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 2 S, the power LCM matrix ((S) e ) 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.