相关论文: Factorization of power GCD matrices and power LCM …

Let $(x, y)$ and $[x, y]$ denote the greatest common divisor and the least common multiple of the integers $x$ and $y$ respectively. We denote by $|T|$ the number of elements of a finite set $T$. Let $a,b$ and $n$ be positive integers and…

Let $a$ and $n$ be positive integers and let $S=\{x_1, \cdots, x_n\}$ be a set of $n$ distinct positive integers. For $x\in S$, one defines $G_{S}(x)=\{d\in S: d<x, d|x \ {\rm and} \ (d|y|x, y\in S)\Rightarrow y\in \{d,x\}\}$. We denote by…

Let $f$ be an arithmetical function. The matrix $[f(i,j)]_{n\times n}$ given by the value of $f$ in greatest common divisor of $(i,j)$, $f\big((i,j)\big)$ as its $i,\; j$ entry is called the greatest common divisor (GCD) matrix. We consider…

Let $S=\{x_1,\ldots, x_n\}$ be a gcd-closed set (i.e. $(x_i,x_j)\in S $ for all $1\le i,j\le n$). In 2002, Hong proposed the divisibility problem of characterizing all gcd-closed sets $S$ with $|S|\ge 4$ such that the GCD matrix $(S)$…

Let $S=\{x_1,x_2,\dots,x_n\}$ be a set of distinct positive integers, and let $f$ be an arithmetical function. The GCD matrix $(S)_f$ on $S$ associated with $f$ is defined as the $n\times n$ matrix having $f$ evaluated at the greatest…

Let $X = (x_1,\dots,x_n)$ be a vector of distinct positive integers. The $n \times n$ matrix $S = S(X) := (\gcd(x_i,x_j))_{i,j=1}^n$, where $\gcd(x_i,x_j)$ denotes the greatest common divisor of $x_i$ and $x_j$, is called the greatest…

In this note we prove the following fact: if finite many elements $p_1,p_2,...,p_n$ of a unique factorization domain are given such that the greatest common divisor of each pair $(p_i,p_j)$ can be expressed as a linear combination of $p_i$…

Let $R$ be a gcd-domain (for example let $R$ be a unique factorization domain), and let $(a_n)_{n\geqslant1}$ be a sequence of nonzero elements in $R$. We prove that $\gcd(a_n,a_m)=a_{\gcd(n,m)}$ for all $n,m\geqslant1$ if and only if…

In 1876 H. J. S. Smith defined an LCM matrix as follows: let S = {x_1, x_2, ..., x_n} be a set of positive integers. The LCM matrix [S] is the n $\times$ n matrix with lcm(x_i , x_j) as its ij entry. During the last 30 years singularity of…

Let $S=\{s_{1},s_{2},\ldots,s_{n}\}$ be an ordered set of $n$ distinct positive integers. The $m$th-order $n$-dimensional tensor $T_{[S]}=(t_{i_{1}i_{2}\ldots i_{m}}),$ where $t_{i_{1}i_{2}\ldots…

Fix a prime $p$ and an integer $m$ with $p> m \geq 2$. Define the family of finite groups \[ G_n :=SL_m (\mathbb{Z}/p^{n}\mathbb{Z}) \] for $n=1,2,... $. We will prove that there exist two positive constants $C$ and $d$ such that for any…

Let $ G $ be a finite group of order $ n$. The strong power graph $\mathcal{P}_s(G) $ of $G$ is the undirected graph whose vertices are the elements of $G$ such that two distinct vertices $a$ and $b$ are adjacent if $a^{{m}_1}$=$b^{{m}_2}$…

Let $f$ be an arithmetical function. The matrix $[f[i,j]]_{n\times n}$ given by the value of $f$ in least common multiple of $[i,j]$, $f\big([i,j]\big)$ as its $i,\; j$ entry is called the least common multiple (LCM) matrix. We consider the…

This article characterizes the rank-one factorization of auto-correlation matrix polynomials. We establish a sufficient and necessary uniqueness condition for uniqueness of the factorization based on the greatest common divisor (GCD) of…

Given a numerical semigroup $S = < a_1, a_2,..., a_t>$ and $s\in S$, we consider the factorization $s = c_1 a_1 + c_2 a_2 +... + c_t a_t$ where $c_i\ge0$. Such a factorization is {\em maximal} if $c_1+c_2+...+c_t$ is a maximum over all such…

The factorizations of the polynomial $X^n-1$ and the cyclotomic polynomial $\Phi_n$ over a finite field $\mathbb F_q$ have been studied for a very long time. Explicit factorizations have been given for the case that $\mathrm{rad}(n)\mid…

Given an integer $k$, define $C_k$ as the set of integers $n > \max(k,0)$ such that $a^{n-k+1} \equiv a \pmod{n}$ holds for all integers $a$. We establish various multiplicative properties of the elements in $C_k$ and give a sufficient…

Let $p$ be a fixed prime, and let $v(a)$ stand for the exponent of $p$ in the prime factorization of the integer $a$. Let $f$ and $g$ be two monic polynomials with integer coefficients and nonzero resultant $r$. Write $S$ for the maximum of…

We prove that for any positive integers $n$ and $k$ such that $n\!\geq\! k\!\geq\! 1$, there exists an $[n,k]$ generalized Reed-Solomon (GRS) code that has a sparsest and balanced generator matrix (SBGM) over any finite field of size…

In this note, we give a necessary and sufficient condition for a matrix A in M to be finitely G-determined, where M is the ring of 2 x 2 matrices whose entries are formal power series over an infinite field, and G is a group acting on M by…