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

摘要

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.