Divisibility among power GCD matrices and among power LCM matrices on two coprime divisor chains

Abstract

Let a, b and h be positive integers and S = {x 1, x 2, …, xh } be a set of h distinct positive integers. The set S is called a divisor chain if there is a permutation σ of {1, …, h} such that x σ(1)|…|x σ(h). We say that the set S consists of two coprime divisor chains if we can partition S as S = S 1 ∪ S 2, where S 1 and S 2 are divisor chains and each element of S 1 is coprime to each element of S 2. The matrix having the a-th power (xi , xj ) a of the GCD (GCD: greatest common divisor) of xi and xj as its i, j-entry is called a-th power GCD matrix defined on S, denoted by (Sa ). Similarly we can define the a-th power least common multiple (LCM) matrix [Sa ]. In this article, we show the following results: assume that S consists of two coprime divisor chains and 1∈S. We first show that if a|b, then the power GCD matrix (Sa ) divides the power GCD matrix (Sb ) in the ring Mh (Z) of h × h matrices over integers. But such factorization should not hold if . Consequently, we show that if a|b, the power LCM matrix [Sa ] divides the power LCM matrix [Sb ] in the ring Mh (Z). Finally we show that if a|b, the power GCD matrix (Sa ) divides the power LCM matrix [Sb ] in the ring Mh (Z). But such results fail to be true if . These results confirm partially Hong's conjectures.