Divisor Chains Research Articles

Divisibility Properties of Power Matrices Associated with Arithmetic Functions on a Divisor Chain

Let [Formula: see text], [Formula: see text] and [Formula: see text] be positive integers with[Formula: see text], [Formula: see text] be an integer-valued arithmetic function, and the set [Formula: see text] of [Formula: see text] distinct positive integers be a divisor chain such that [Formula: see text]. We first show that the matrix [Formula: see text] having [Formula: see text] evaluated at the [Formula: see text]th power [Formula: see text] of the greatest common divisor of [Formula: see text] and [Formula: see text] as its [Formula: see text]-entry divides the GCD matrix [Formula: see text] in the ring [Formula: see text] of [Formula: see text] matrices over integers if and only if [Formula: see text] and [Formula: see text] divides [Formula: see text] for any integer [Formula: see text] with [Formula: see text]. Consequently, we show that the matrix [Formula: see text] having [Formula: see text] evaluated at the [Formula: see text]th power [Formula: see text] of the least common multiple of [Formula: see text] and [Formula: see text] as its [Formula: see text]-entry divides the matrix [Formula: see text] in the ring [Formula: see text] if and only if [Formula: see text] and [Formula: see text] divides [Formula: see text] for any integer [Formula: see text] with[Formula: see text]. Finally, we prove that the matrix [Formula: see text] divides the matrix [Formula: see text] in the ring [Formula: see text] if and only if [Formula: see text] and [Formula: see text] for any integer [Formula: see text] with [Formula: see text]. Our results extend and strengthen the theorems of Hong obtained in 2008.

Divisibility among determinants of power matrices associated with integer-valued arithmetic functions

Let $a, b$ and $n$ be positive integers and $S = \left\{ {x_1, ..., x_n} \right\}$ be a set of $n$ distinct positive integers. The set $S$ is called a divisor chain if there is a permutation $\sigma $ of $\{1, ..., n\}$ such that $x_{\sigma (1)}|...|x_{\sigma (n)}$. We say that the set $S$ consists of two coprime divisor chains if we can partition $S$ as $S = S_1\cup S_2$, where $S_1$ and $S_2$ are divisor chains and each element of $S_1$ is coprime to each element of $S_2$. For any arithmetic function $f$, we define the function $f^a$ for any positive integer $x$ by $f^a(x): = (f(x))^a$. The matrix $(f^a(S))$ is the $n\times n$ matrix having $f^a$ evaluated at the the greatest common divisor of $x_{i}$ and $x_{j}$ as its $(i, j)$-entry and the matrix $(f^a[S])$ is the $n\times n$ matrix having $f^a$ evaluated at the least common multiple of $x_i$ and $x_j$ as its $(i, j)$-entry. In this paper, when $f$ is an integer-valued arithmetic function and $S$ consists of two coprime divisor chains with $1 \not\in S$, we establish the divisibility theorems between the determinants of the power matrices $(f^a(S))$ and $(f^b(S))$, between the determinants of the power matrices $(f^a[S])$ and $(f^b[S])$ and between the determinants of the power matrices $(f^a(S))$ and $(f^b[S])$. Our results extend Hong's theorem obtained in 2003 and the theorem of Tan, Lin and Liu gotten in 2011.

Divisibility among power matrices associated with arithmetic functions on finitely many quasi-coprime divisor chains

Let a, b and h be positive integers and be a set of h distinct positive integers. The set S is called a divisor chain if there is a permutation on such that . We say that the set S consists of finitely many quasi-coprime divisor chains if we can partition S as , where is an integer and all are divisor chains such that for any . For any arithmetic function f, define the function for any positive integer n by . The matrix is the matrix having evaluated at the the greatest common divisor of and as its (i, j)-entry and the matrix is the matrix having evaluated at the least common multiple of and as its (i, j) entry. In this paper, when f is an integer-valued arithmetic function and S consists of finitely many quasi-coprime divisor chains, we establish the divisibility theorems between the power matrices and , and between the power matrices and in the ring of matrices over integers. This extends Hong’s theorem obtained in 2008 and the theorem of Tan and Li gotten in 2013.

Determinants of matrices associated with arithmetic functions on finitely many quasi-coprime divisor chains

Let S={x1,…,xn} be a set of n distinct positive integers and f be an arithmetic function. We use f(S)=f(xi,xj) (resp. f[S]=f[xi,xj]) to denote the n×n matrix having f evaluated at the greatest common divisor (resp. the least common multiple) of xi and xj as its i,j-entry. The set S is called a divisor chain if there is a permutation σ of {1,…,n} such that xσ(1)|…|xσ(n). If S can be partitioned as S=⋃i=1kSi with all Si(1⩽i⩽k) being divisor chains and (max(Si), max(Sj))=gcd(S) for 1⩽i≠j⩽k, then we say that S consists of finitely many quasi-coprime divisor chains. In this paper, we introduce a new method to give the formulas for the determinants of the matrices (f(S)) and (f[S]) on finitely many quasi-coprime divisor chains S. We show also that det(f(S))|det(f[S]) holds under some natural conditions. These extend the results obtained by Tan and Lin (2010) and Tan et al. (2013), respectively.

Multiple divisor chains and determinants of matrices associated with completely even functions (mod r)

Let f and g be arithmetic functions and be a set of n distinct positive integers. Let and denote the matrices having f evaluated at the greatest common divisor of and and the least common multiple of and as their -entry, respectively. We say that S is a divisor chain if there is a permutation such that . If S can be partitioned as with k being a positive integer and all being divisor chains and each element of being coprime to each element of for all integers i and j with , then S is called multiple coprime divisor chains. In this paper, we give the formulae for the determinants of the matrices (()) and ([]) on the multiple coprime divisor chains S. Consequently, we show that if S consists of at most two coprime divisor chains and f is multiplicative with () being a non-zero integer for all , then . But, such result fails to be true if S consists of at least three coprime divisor chains. Finally, some examples are given to illustrate the validity of the main results.

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

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.

Divisibility among power GCD matrices and among power LCM matrices on finitely many coprime divisor chains

Let a,b and h be positive integers and S={x1,...,xh} be a set of h distinct positive integers. The set S is called a divisor chain if there is a permutation σ on {1,...,h} such that xσ(1)|...|xσ(h). We say that the set S consists of finitely many coprime divisor chains if there is a positive integer k such that we can partition S as S=S1∪...∪Sk, where all the Si are divisor chains and each element of Si is coprime to each element of Sj for 1⩽i≠j⩽k. The matrix having the ath power (xi,xj)a of the greatest common divisor of xi and xj as its (i,j)-entry is called ath power greatest common divisor (GCD) matrix defined on S, denoted by (Sa). Similarly we can define the ath power LCM matrix Sa. In this paper, we show that if a|b and S consists of finitely many coprime divisor chains with 1∈S, then in the ring Mh(Z) of h×h matrices over integers, we have (Sa)|(Sb),[Sa]|[Sb] and (Sa)|[Sb]. But such results fail to be true if a|b. These results confirm partially Hong’s conjectures raised in 2008.

Determinants of Smith matrices on three coprime divisor chains and divisibility

Let a, b and h be positive integers and S = {x 1, … , x h } 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 three coprime divisor chains if we can partition S 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. 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 ath power greatest common divisor (GCD) matrix defined on S, denoted by (S a ). Similarly we can define the ath power LCM matrix [S a ]. In this article, we first obtain the formulae for determinants of power GCD matrices and power LCM matrices defined on three coprime divisor chains. Consequently, we establish several divisibility theorems among the determinants of these power matrices: if S consists of three coprime divisor chains and a|b, then under some natural conditions, we have det(S a )|det(S b ), det[S a ] |det[S b ] and det(S a )|det[S b ]. We also give an example which shows that such divisibility results are not true if those natural conditions are not satisfied.

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

Let a, b and h be positive integers and S = {x 1, … , x h } 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 ath power (x i , x j ) a of the greatest common divisor (GCD) of x i and x j as its (i,j)-entry is called the ath power GCD matrix defined on S, denoted by (S a ). Similarly, we can define the ath power least common multiple (LCM) matrix [S a ]. In the first paper of the series, Tan [Q. Tan, Divisibility among power GCD matrices and among power LCM matrices on two coprime divisor chains, Linear Multilinear Algebra 58 (2010), pp. 659--671] showed that if S consists of two coprime divisor chains and 1 ∈ S and a|b, then (S a )|(S b ), [S a ]|[S b ] and (S a )|[S b ] hold in the ring M h (Z) of h × h matrices over integers. But such factorizations need not hold if a ∤ b. In this second paper of the series, we assume that S consists of two coprime divisor chains and 1 ∉ S. We show the following results: (i) If , then , and . (ii) If a|b, then det(S a ) | det(S b ), det[S a ] | det[S b ] and det(S a ) | det[S b ]. (iii) If a|b, then (S a ) | (S b ), [S a ] | [S b ] and (S a ) | [S b ] hold in the ring M h (Z) if and only if both and are integers, where S = S 1 ∪ S 2 with S 1 and S 2 divisor chains and x = min(S 1) and y = min(S 2). Our results extend Hong’s results and complement Tan’s results.

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

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.

Divisibility of determinants of power GCD matrices and power LCM matrices on finitely many quasi-coprime divisor chains

Let a, n ⩾ 1 be integers and S = { x 1, … , x n } be a set of n distinct positive integers. 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 ath power greatest common divisor (GCD) matrix defined on S, denoted by ( S a ). Similarly we can define the ath power LCM matrix [ S a ]. We say that the set S consists of finitely many quasi-coprime divisor chains if we can partition S as S = S 1 ∪ ⋯ ∪ S k , where k ⩾ 1 is an integer and all S i (1 ⩽ i ⩽ k) are divisor chains such that (max( S i ), max( S j )) = gcd( S) for 1 ⩽ i ≠ j ⩽ k. In this paper, we first obtain formulae of determinants of power GCD matrices ( S a ) and power LCM matrices [ S a ] on the set S consisting of finitely many quasi-coprime divisor chains with gcd( S) ∈ S. Using these results, we then show that det( S a )∣det( S b ), det[ S a ]∣det[ S b ] and det( S a )∣det[ S b ] if a∣ b and S consists of finitely many quasi-coprime divisor chains with gcd( S) ∈ S. But such factorizations fail to be true if such divisor chains are not quasi-coprime.

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

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.

Divisibility properties of power GCD matrices and power LCM matrices

Let a , b and n be positive integers and the set S = { x 1 , … , x n } of n distinct positive integers be a divisor chain (i.e. there exists a permutation σ on { 1 , … , n } such that x σ ( 1 ) | … | x σ ( n ) ). In this paper, we show that if a | b , then the a th power GCD matrix ( S a ) having the a th power ( x i , x j ) a of the greatest common divisor of x i and x j as its i , j -entry divides the b th power GCD matrix ( S b ) in the ring M n ( Z ) of n × n matrices over integers. We show also that if a ∤ b and n ⩾ 2 , then the a th power GCD matrix ( S a ) does not divide the b th power GCD matrix ( S b ) in the ring M n ( Z ) . Similar results are also established for the power LCM matrices.

Gauge functions and the divisor chain condition

A gauge function on a commutative cancellative monoid M is a map p : M?Z+(the set of nonnegative) integers such that for all a,b e?M, p(ab)≥ p(a)+p(b) and p(a) = 0 if and only if a is a unit. In [3], it is shown by V. Srinivasan and H. Shaing that if the monoid of nonzero elements of an integral domain A in which finitely‐generated ideals are principal admits a gauge function, then A is a principal ideal domain. In this paper I show that M admits a gauge function if and only if for each nonunit a e M there is a positive integer N = N(a) such that α is a product of no more than N(a) irreducibles. An example is given to show that this latter condition is stronger than the divisor chain condition defined in [2, 2.14].