Notes on Hong's conjectures of real number power LCM matrices

Abstract

Let e be a real number and S = { x 1 , … , x n } be a set of n distinct positive integers. The set S is said to be gcd-closed (respectively lcm-closed) if ( x i , x j ) ∈ S (respectively [ x i , x j ] ∈ S ) for all 1 ⩽ i , j ⩽ n . The matrix having eth 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 eth power least common multiple (LCM) matrix, denoted by ( [ x i , x j ] e ) (or abbreviated by ( [ S ] e ) ). In this paper, we show that for any real number e ⩾ 1 and n ⩽ 7 , the power LCM matrix ( [ x i , x j ] e ) defined on any gcd-closed (respectively lcm-closed) set S = { x 1 , … , x n } is nonsingular. This confirms partially two conjectures raised by Hong in [S. Hong, Nonsingularity of matrices associated with classes of arithmetical functions, J. Algebra 281 (2004) 1–14]. Similar results are established for reciprocal real number power GCD matrices.