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

摘要

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.