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.

Keywords:

• divisibility
• three coprime divisor chains
• greatest-type divisor
• power GCD matrix
• power LCM matrix

AMS Subject Classifications:

• 11C20
• 11A05
• 15A36

Acknowledgements

The authors would like to thank the anonymous referees and the editor for their helpful suggestions and valuable comments. This research was supported partially by the PhD Programmes Foundation of the Ministry of Education of China, Grant No. 201001811107, and by the Fundamental Research Funds for the Central Universities, Grant No. XDJK2010C058.