Non-divisibility of LCM Matrices by GCD Matrices on GCD-closed Sets

TL;DR

This paper investigates the divisibility of LCM matrices by GCD matrices on gcd-closed sets, providing theoretical evidence supporting Zhao's conjecture and establishing conditions for divisibility in small sets.

Contribution

It presents the first theoretical examples where LCM matrices are not divisible by GCD matrices, and characterizes divisibility conditions for small gcd-closed sets, advancing understanding of the divisibility problem.

Findings

Identified gcd-closed sets where divisibility does not hold

Provided necessary and sufficient conditions for divisibility in sets with up to 8 elements

Proposed a new conjecture generalizing Zhao's conjecture

Abstract

In this paper, we consider the divisibility problem of LCM matrices by GCD matrices in the ring $M_n(\mathbb{Z})$ proposed by Hong in 2002 and in particular a conjecture concerning the divisibility problem raised by Zhao in 2014. We present some certain gcd-closed sets on which the LCM matrix is not divisible by the GCD matrix in the ring $M_n(\mathbb{Z})$. This could be the first theoretical evidence that Zhao's conjecture might be true. Furthermore, we give the necessary and sufficient conditions on the gcd-closed set $S$ with $|S|\leq 8$ such that the GCD matrix divides the LCM matrix in the ring $M_n(\mathbb{Z})$ and hence we partially solve Hong's problem. Finally, we conclude with a new conjecture that can be thought as a generalization of Zhao's conjecture.