Determinants and divisibility of power GCD and power LCM matrices on finitely many coprime divisor chains

Abstract

Let a,b and h be positive integers and S={x1,…,xh} be a set of h distinct positive integers. The h×h matrix (Sa)=((xi,xj)a), having the ath power (xi,xj)a of the greatest common divisor of xi and xj as its (i,j)-entry, is called the ath power GCD matrix on S. The ath power LCM matrix on S can be defined similarly. In this paper, we first obtain the formulae for determinants of power GCD and power LCM matrices on the set S consisting of finitely many coprime divisor chains (i.e., there is a positive integer k such that we can partition S as S=S1∪⋯∪Sk, where Si and Sj are divisor chains and each element of Si is coprime to each element of Sj for any 1⩽i≠j⩽k). Consequently, we show that if S consists of finitely many coprime divisor chains, then under some natural conditions, we have det(Sa)|det(Sb),det[Sa]|det[Sb] and det(Sa)|det[Sb]. Our results extend Hong’s 2008 theorem and complements Tan–Lin 2010 theorem.