Gcd-closed Set Research Articles

On Mersenne Power GCD Matrices

This paper explores the 𝑛 × 𝑛 Mersenne power GCD matrices defined on sets of positive integers, focusing on factor-closed and gcd-closed sets. By employing the form 𝑓(𝑡𝑖 ,𝑡𝑗) = 2 (𝑡𝑖 ,𝑡𝑗 ) − 1, we investigate the 𝑟 𝑡ℎ power Mersenne GCD matrix (𝑀𝑟 ) and provide comprehensive insights into its factorizations, determinants, reciprocals, and inverses. Building upon previous research, particularly Chun's work on power GCD matrices, we extend the analysis to Mersenne numbers, offering a thorough understanding of their properties. The study contributes to the broader understanding of arithmetic functions and their applications in matrix theory.

Divisibility among power matrices associated with multiplicative functions

Let a, b and n be positive integers and let S = { x 1 , … , x n } be a set of n distinct positive integers. For x ∈ S , one defines G S ( x ) = { y ∈ S : y < x , y | x and ( y | z | x , z ∈ S ) ⇒ z ∈ { y , x } } . For any arithmetic function f and any positive integer x, we define the power arithmetic function f a by f a ( x ) = ( f ( x ) ) a . We denote by ( f a [ S ] ) the n × n power matrix having f a evaluated at the least common multiple of x i and x j as its ( i , j ) -entry. Denote by lcm ( S ) the least common multiple of all the elements of S and by | T | the number of elements of any finite set T. In this paper, we show that if a | b , S is gcd closed (i.e. gcd ( x i , x j ) ∈ S for all integers i and j with 1 ≤ i , j ≤ n ) with max x ∈ S { | G S ( x ) | } = 1 and f is a multiplicative function such that ( f a ] S ] ) is nonsingular and ( f ∗ μ ) ( d ) ∈ Z whenever d | lcm ( S ) and f ( y ) | f ( x ) whenever y | x and y , x ∈ S , where f ∗ μ is the Dirichlet convolution of f and the Möbius function μ, then the ath power matrix ( f a [ S ] ) divides the bth power matrix ( f b [ S ] ) in the ring of n × n matrices over the integers. Our result extend a theorem of S.F. Hong [Divisibility properties of power GCD matrices and power LCM matrices. Linear Algebra Appl. 2008;428:1001–1008] and that of G.Y. Zhu and M. Li [On the divisibility among power LCM matrices on gcd-closed sets. Bull Aust Math Soc. 2023;107:31–39].

Gcd-closed sets and divisibility of Smith matrices

Let n be a positive integer and S={x1,...,xn} a set of n distinct positive integers. In 1992, Bourque and Ligh showed that if S is factor closed (i.e. d∈S if d|x for any x∈S), then the GCD matrix (S) divides the LCM matrix [S] in the ring of n×n matrices over the integers. In 2002, Hong showed that such factorization holds for any gcd-closed set S (i.e. (xi,xj)∈S for all 1≤i,j≤n) with |S|≤3. But for any integer n≥4, there is a gcd-closed set S with |S|=n such that (S)∤[S]. Meanwhile, Hong proposed the problem of characterizing all gcd-closed sets S with |S|≥4 such that (S)|[S]. For x∈S, let GS(x):={d∈S|d<x,d|xand(d|y|x,y∈S)⇒y∈{d,x}}. In 2009, Feng, Hong and Zhao answered this problem when maxx∈S{|GS(x)|}≤2. Zhao in 2014 and Altinisik, Yildiz and Keskin in 2017 characterized all gcd-closed sets S such that (S)|[S] for the case 4≤|S|≤7 and |S|=8, respectively. In this paper, we introduce a new method to study Hong's problem. We obtain a necessary and sufficient condition on the gcd-closed set S with maxx∈S{|GS(x)|}=3 such that (S)|[S]. This solves partially Hong's problem.

On the divisibility among power GCD and power LCM matrices on gcd-closed sets

Let [Formula: see text] and [Formula: see text] be positive integers and let [Formula: see text] be a set of [Formula: see text] distinct positive integers. For [Formula: see text], one defines [Formula: see text]. We denote by [Formula: see text] (respectively, [Formula: see text]) the [Formula: see text] matrix having the [Formula: see text]th power of the greatest common divisor (respectively, the least common multiple) of [Formula: see text] and [Formula: see text] as its [Formula: see text]-entry. In this paper, we show that for arbitrary positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], the [Formula: see text]th power matrices [Formula: see text] and [Formula: see text] are both divisible by the [Formula: see text]th power matrix [Formula: see text] if [Formula: see text] is a gcd-closed set (i.e. [Formula: see text] for all integers [Formula: see text] and [Formula: see text] with [Formula: see text]) such that [Formula: see text]. This confirms two conjectures of Shaofang Hong proposed in 2008.

Power GCD and power LCM matrices defined on GCD-closed sets over unique factorization domains

Power GCD and power LCM matrices defined on GCD-closed sets over unique factorization domains

A lattice-theoretic approach to the Bourque–Ligh conjecture

ABSTRACTThe Bourque–Ligh conjecture states that if is a gcd-closed set of positive integers with distinct elements, then the LCM matrix is invertible. It is known that this conjecture holds for but does not generally hold for . In this paper, we study the invertibility of matrices in a more general matrix class, join matrices. At the same time, we provide a lattice-theoretic explanation for this solution of the Bourque–Ligh conjecture. In fact, let be a lattice, let be a subset of P and let be a function. We study under which conditions the join matrix on S with respect to f is invertible on a meet closed set S (i.e. .

Non-divisibility of LCM matrices by GCD matrices on gcd-closed sets

In this paper, we consider the divisibility problem of LCM matrices by GCD matrices in the ring Mn(Z) proposed by Shaofang Hong in 2002 and in particular a conjecture concerning the divisibility problem raised by Jianrong Zhao in 2014. We present some certain gcd-closed sets on which the LCM matrix is not divisible by the GCD matrix in the ring Mn(Z). This could be the first theoretical evidence that Zhao's conjecture might be true. Furthermore, we give the necessary and sufficient conditions on the gcd-closed set S with |S|≤8 such that the GCD matrix divides the LCM matrix in the ring Mn(Z) and hence we partially solve Hong's problem. Finally, we conclude with a new conjecture that can be thought as a generalization of Zhao's conjecture.

Multiple gcd-closed sets and determinants of matrices associated with arithmetic functions

Abstract Let f be an arithmetic function and S = {x 1, …, xn } be a set of n distinct positive integers. By (f(xi , xj )) (resp. (f[xi , xj ])) we denote the n × n matrix having f evaluated at the greatest common divisor (xi , xj ) (resp. the least common multiple [xi , xj ]) of x, and xj as its (i, j)-entry, respectively. The set S is said to be gcd closed if (xi , xj ) ∈ S for 1 ≤ i, j ≤ n. In this paper, we give formulas for the determinants of the matrices (f(xi , xj )) and (f[xi , xj ]) if S consists of multiple coprime gcd-closed sets (i.e., S equals the union of S 1, …, Sk with k ≥ 1 being an integer and S 1, …, Sk being gcd-closed sets such that (lcm(Si ), lcm(Sj )) = 1 for all 1 ≤ i ≠ j ≤ k). This extends the Bourque-Ligh, Hong’s and the Hong-Loewy formulas obtained in 1993, 2002 and 2011, respectively. It also generalizes the famous Smith’s determinant.

Hyperdeterminants associated with multiple even functions

In this paper, we introduce a new method to give the formulae for the hyperdeterminants of higher-dimensional matrices associated with multiple even function (modr) on the gcd-closed sets. Our result generalizes the result of Yamasaki obtained in 2010 and also extends the results of Haukkanen and Hong obtained in 1992 and 2002, respectively.

New results on nonsingular power LCM matrices

Let e and n be positive integers and S = {x1,...,xn} be a set of n distinct positive integers. The n × n matrix having eth power (xi,xj) e of the least common multiple of xi and xj as its (i,j)-entry is called the eth power least common multiple (LCM) matrix on S, denoted by ((S) e ). The set S is said to be gcd closed (respectively, lcm closed) if (xi,xj) 2 S (respectively, (xi,xj) 2 S) for all 1 � i, jn. In 2004, Shaofang Hong showed that the power LCM matrix ((S) e ) is nonsingular if S is a gcd-closed set such that each element of S holds no more than two distinct two prime factors. In this paper, this result is improved by showing that if S is a gcd-closed set such that every element of S contains at most two distinct prime factors or is of the form p l qr with p, q and r being distinct primes and 1 � l � 4 being an integer, then except for the case that e = 1 and 270, 520, 810, 1040 2 S, the power LCM matrix ((S) e ) on S is nonsingular. This gives an evidence to a conjecture of Hong raised in 2002. For the lcm-closed case, similar results are established.

Divisibility of power LCM matrices by power GCD matrices on gcd-closed sets

Let and be positive integers and a set of distinct positive integers. The matrix whose -entry is th power of the greatest common divisor (GCD) of and is called the th power GCD matrix on , denoted by . Similarly, we can define th power least common multiple (LCM) matrix . The set is said to be gcd closed if for all . In this paper, we give the necessary and sufficient conditions on the gcd-closed set with such that the power GCD matrix divides the power LCM matrix in the ring of the matrices over the integers. This solves partially an open problem raised by Shaofang Hong in 2002.

Divisibility of matrices associated with multiplicative functions

Let S={x1,…,xn} be a set of n distinct positive integers. For x,y∈S and y<x, we say the y is a greatest-type divisor of x in S if y∣x and it can be deduced that z=y from y∣z,z∣x,z<x and z∈S. For x∈S, let GS(x) denote the set of all greatest-type divisors of x in S. For any arithmetic function f, let (f(xi,xj)) denote the n×n matrix having f evaluated at the greatest common divisor (xi,xj) of xi and xj as its i,j-entry and let (f[xi,xj]) denote the n×n matrix having f evaluated at the least common multiple [xi,xj] of xi and xj as its i,j-entry. In this paper, we assume that S is a gcd-closed set and maxx∈S{|GS(x)|}=1. We show that if f is a multiplicative function such that (f∗μ)(d)∈Z whenever d|lcm(S) and f(a)|f(b) whenever a|b and a,b∈S and (f(xi,xj)) is nonsingular, then the matrix (f(xi,xj)) divides the matrix (f[xi,xj]) in the ring Mn(Z) of n×n matrices over the integers. As a consequence, we show that (f(xi,xj)) divides (f[xi,xj]) in the ring Mn(Z) if (f∗μ)(d)∈Z whenever d|lcm(S) and f is a completely multiplicative function such that (f(xi,xj)) is nonsingular. This confirms a conjecture of Hong raised in 2004.

Divisibility properties of power LCM matrices by power GCD matrices on gcd-closed sets

Let e and n be positive integers and S = { x 1 , … , x n } a set of n distinct positive integers. For x ∈ S , define G S ( x ) ≔ { d ∈ S | d < x , d | x and ( d | y | x , y ∈ S ) ⇒ y ∈ { d , x } } . The n × n matrix whose ( i , j ) -entry is the e th power ( x i , x j ) e of the greatest common divisor of x i and x j is called the e th power GCD matrix on S , denoted by ( S e ) . Similarly we can define the e th power LCM matrix [ S e ] . Bourque and Ligh showed that ( S ) ∣ [ S ] holds in the ring of n × n matrices over the integers if S is factor closed. Hong showed that for any gcd-closed set S with | S | ≤ 3 , ( S ) ∣ [ S ] . Meanwhile Hong proved that there is a gcd-closed set S with max x ∈ S { | G S ( x ) | } = 2 such that ( S ) ∤ [ S ] . In this paper, we introduce a new method to study systematically the divisibility for the case max x ∈ S { | G S ( x ) | } ≤ 2 . We give a new proof of Hong’s conjecture and obtain necessary and sufficient conditions on the gcd-closed set S with max x ∈ S { | G S ( x ) | } = 2 such that ( S e ) | [ S e ] . This partially solves an open question raised by Hong. Furthermore, we show that such factorization holds if S is a gcd-closed set such that each element is a prime power or the product of two distinct primes, and in particular if S is a gcd-closed set with every element less than 12.

THERE DOES NOT EXIST AN ODD SINGULAR NUMBER OF THE FORM plqr

Motivated by his solution to the Bourque-Ligh conjecture on the nonsingularity of the least common multiple matrix defined on the gcd-closed set, Hong introduced the concept of primitive singular number. Meanwhile Hong proved that there does not exist a singular number with no more than two distinct prime factors. Hong and Shum as well Sun showed in 2006 that there are even primitive singular numbers of the form plqr, where p, q, r are distinct primes and l is a positive integer. In this paper, we show that there does not exist an odd singular number of the form plqr. This improves a result obtained by Hong and Shum as well Sun and also confirms partially a conjecture of Hong.

On Hong’s conjecture for power LCM matrices

A set S={x1,...,xn} of n distinct positive integers is said to be gcd-closed if (xi, xj) ∈ S for all 1 ⩽ i, j ⩽ n. Shaofang Hong conjectured in 2002 that for a given positive integer t there is a positive integer k(t) depending only on t, such that if n ⩽ k(t), then the power LCM matrix ([xi, xj]t) defined on any gcd-closed set S={x1,...,xn} is nonsingular, but for n ⩾ k(t) + 1, there exists a gcd-closed set S={x1,...,xn} such that the power LCM matrix ([xi, xj]t) on S is singular. In 1996, Hong proved k(1) = 7 and noted k(t) ⩾ 7 for all t ⩾ 2. This paper develops Hong’s method and provides a new idea to calculate the determinant of the LCM matrix on a gcd-closed set and proves that k(t) ⩾ 8 for all t ⩾ 2. We further prove that k(t) ⩾ 9 iff a special Diophantine equation, which we call the LCM equation, has no t-th power solution and conjecture that k(t) = 8 for all t ⩾ 2, namely, the LCM equation has t-th power solution for all t ⩾ 2.

On Nonsingular Power LCM Matrices

Let e ≥ 1 be an integer and S={x1,…,xn} a set of n distinct positive integers. The matrix ([xi, xj]e) having the power [xi, xj]e of the least common multiple of xi and xj as its (i, j)-entry is called the power least common multiple (LCM) matrix defined on S. The set S is called gcd-closed if (xi,xj) ∈ S for 1≤ i, j≤ n. Hong in 2004 showed that if the set S is gcd-closed such that every element of S has at most two distinct prime factors, then the power LCM matrix on S is nonsingular. In this paper, we use Hong's method developed in his previous papers to consider the next case. We prove that if every element of an arbitrary gcd-closed set S is of the form pqr, or p2qr, or p3qr, where p, q and r are distinct primes, then except for the case e=1 and 270, 520 ∈ S, the power LCM matrix on S is nonsingular. We also show that if S is a gcd-closed set satisfying xi&lt; 180 for all 1≤ i≤ n, then the power LCM matrix on S is nonsingular. This proves that 180 is the least primitive singular number. For the lcm-closed case, we establish similar results.

Nonsingularity of least common multiple matrices on gcd-closed sets

Let n be a positive integer. Let S = { x 1 , … , x n } be a set of n distinct positive integers. The least common multiple (LCM) matrix on S, denoted by [ S ] , is defined to be the n × n matrix whose ( i , j ) -entry is the least common multiple [ x i , x j ] of x i and x j . The set S is said to be gcd-closed if for any x i , x j ∈ S , ( x i , x j ) ∈ S . For an integer m > 1 , let ω ( m ) denote the number of distinct prime factors of m. Define ω ( 1 ) = 0 . In 1997, Qi Sun conjectured that if S is a gcd-closed set satisfying max x ∈ S { ω ( x ) } ⩽ 2 , then the LCM matrix [ S ] is nonsingular. In this paper, we settle completely Sun's conjecture. We show the following result: (i). If S is a gcd-closed set satisfying max x ∈ S { ω ( x ) } ⩽ 2 , then the LCM matrix [ S ] is nonsingular. Namely, Sun's conjecture is true; (ii). For each integer r ⩾ 3 , there exists a gcd-closed set S satisfying max x ∈ S { ω ( x ) } = r , such that the LCM matrix [ S ] is singular.

Nonsingularity of matrices associated with classes of arithmetical functions

Let S = { x 1 , … , x n } be a set of n distinct positive integers. Let f be an arithmetical function. Let [ f ( x i , x j ) ] denote the n × n matrix having f evaluated at the greatest common divisor ( x i , x j ) of x i and x j as its i , j -entry and ( f [ x i , x j ] ) denote the n × n matrix having f evaluated at the least common multiple [ x i , x j ] of x i and x j as its i , j -entry. The set S is said to be gcd-closed if ( x i , x j ) ∈ S for all 1 ⩽ i , j ⩽ n . For an integer x, let ν ( x ) denote the number of distinct prime factors of x. In this paper, by using the concept of greatest-type divisor introduced by S. Hong in [Adv. Math. (China) 25 (1996) 566–568; J. Algebra 218 (1999) 216–228], we obtain a new reduced formula for det f [ ( x i , x j ) ] if S is gcd-closed. Then we show that if S = { x 1 , … , x n } is a gcd-closed set satisfying max x ∈ S { ν ( x ) } ⩽ 2 , and if f is a strictly increasing (respectively decreasing) completely multiplicative function, or if f is a strictly decreasing (respectively increasing) completely multiplicative function satisfying 0 < f ( p ) ⩽ 1 p (respectively f ( p ) ⩾ p ) for any prime p, then the matrix [ f ( x i , x j ) ] (respectively ( f [ x i , x j ] ) ) defined on S is nonsingular. As a corollary, we show the following interesting result: The LCM matrix ( [ x i , x j ] ) defined on a gcd-closed set is nonsingular if max x ∈ S { ν ( x ) } ⩽ 2 .

The GCD-Reciprocal LCM Matrices on GCD-Closed Sets

We have given a structure theorem for the GCD-Reciprocal LCM matrix and then we have calculated the value of the determinant of the GCD-Reciprocal LCM matrix. We have obtained formula for the determinant and the inverse of the GCD-Reciprocal LCM matrix defined on gcd-closed sets.

On the factorization of LCM matrices on gcd-closed sets

Let S={ x 1,…, x n } be a set of n distinct positive integers. The matrix having the greatest common divisor (GCD) ( x i , x j ) of x i and x j as its i, j-entry is called the greatest common divisor matrix, denoted by ( S) n . The matrix having the least common multiple (LCM) [ x i , x j ] of x i and x j as its i, j-entry is called the least common multiple matrix, denoted by [ S] n . The set is said to be gcd-closed if ( x i , x j )∈ S for all 1⩽ i, j⩽ n. In this paper we show that if n⩽3, then for any gcd-closed set S={ x 1,…, x n }, the GCD matrix on S divides the LCM matrix on S in the ring M n( Z ) of n× n matrices over the integers. For n⩾4, there exists a gcd-closed set S={ x 1,…, x n } such that the GCD matrix on S does not divide the LCM matrix on S in the ring M n( Z ) . This solves a conjecture raised by the author in 1998.