Divisibility properties of power LCM matrices by power GCD matrices on gcd-closed sets

Abstract

Let e and n be positive integers and S = { x 1 , … , x n } a set of n distinct positive integers. For x ∈ S , define G S ( x ) ≔ { d ∈ S | d < x , d | x and ( d | y | x , y ∈ S ) ⇒ y ∈ { d , x } } . The n × n matrix whose ( i , j ) -entry is the e th power ( x i , x j ) e of the greatest common divisor of x i and x j is called the e th power GCD matrix on S , denoted by ( S e ) . Similarly we can define the e th power LCM matrix [ S e ] . Bourque and Ligh showed that ( S ) ∣ [ S ] holds in the ring of n × n matrices over the integers if S is factor closed. Hong showed that for any gcd-closed set S with | S | ≤ 3 , ( S ) ∣ [ S ] . Meanwhile Hong proved that there is a gcd-closed set S with max x ∈ S { | G S ( x ) | } = 2 such that ( S ) ∤ [ S ] . In this paper, we introduce a new method to study systematically the divisibility for the case max x ∈ S { | G S ( x ) | } ≤ 2 . We give a new proof of Hong’s conjecture and obtain necessary and sufficient conditions on the gcd-closed set S with max x ∈ S { | G S ( x ) | } = 2 such that ( S e ) | [ S e ] . This partially solves an open question raised by Hong. Furthermore, we show that such factorization holds if S is a gcd-closed set such that each element is a prime power or the product of two distinct primes, and in particular if S is a gcd-closed set with every element less than 12.