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

Abstract

Let h be a positive integer and S = {x 1, … , x h } be a set of h distinct positive integers. We say that the set S is a divisor chain if x σ(1) ∣ … ∣ x σ(h) for a permutation σ of {1, … , h}. If the set S can be partitioned as S = S 1 ∪ S 2 ∪ S 3, where S 1, S 2 and S 3 are divisor chains and each element of S i is coprime to each element of S j for all 1 ≤ i < j ≤ 3, then we say that the set S consists of three coprime divisor chains. The matrix having the ath power (x i , x j ) a of the greatest common divisor of x i and x j as its i, j-entry is called the ath power greatest common divison (GCD) matrix on S, denoted by (S a ). The ath power least common multiple (LCM) matrix [S a ] can be defined similarly. In this article, let a and b be positive integers and let S consist of three coprime divisor chains with 1 ∈ S. We show that if a ∣ b, then the ath power GCD matrix (S a ) (resp., the ath power LCM matrix [S a ]) divides the bth power GCD matrix (S b ) (resp., the bth power LCM matrix [S b ]) in the ring M h (Z) of h × h matrices over integers. We also show that the ath power GCD matrix (S a ) divides the bth power LCM matrix [S b ] in the ring M h (Z) if a ∣ b. However, if a ∤ b, then such factorizations are not true. Our results extend Hong's and Tan's theorems and also provide further evidences to the conjectures of Hong raised in 2008.