On Nonsingular Power LCM Matrices

2006 ◽  
Vol 13 (04) ◽  
pp. 689-704 ◽  
Author(s):  
Shaofang Hong ◽  
K. P. Shum ◽  
Qi Sun
Keyword(s):  
Singular Number ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Closed Set ◽  
Prime Factors ◽  
The Matrix ◽  
Lcm Matrix ◽  
Positive Integers ◽  
Power Lcm Matrix

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< 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.

scholarly journals ASYMPTOTIC BEHAVIOR OF EIGENVALUES OF RECIPROCAL POWER LCM MATRICES

2008 ◽  
Vol 50 (1) ◽  
pp. 163-174 ◽  
Author(s):  
SHAOFANG HONG ◽  
K. S. ENOCH LEE
Keyword(s):  
Asymptotic Behavior ◽  
Real Number ◽  
Infinite Sequence ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Lcm Matrix ◽  
Positive Integers ◽  
Power Lcm Matrix

AbstractLet$\{x_i\}_{i=1}^{\infty}$be an arbitrary strictly increasing infinite sequence of positive integers. For an integern≥1, let$S_n=\{x_1, {\ldots}\, x_n\}$. Letr>0 be a real number andq≥ 1 a given integer. Let$\lambda _n^{(1)}\, {\le}\, {\ldots}\, {\le}\, \lambda _n^{(n)}$be the eigenvalues of the reciprocal power LCM matrix$(\frac{1}{[x_i, x_j]^r})$having the reciprocal power${1\over {[x_i, x_j]^r}}$of the least common multiple ofxiandxjas itsi,j-entry. We show that the sequence$\{\lambda _n^{(q)}\}_{n=q}^{\infty}$converges and${\rm lim}_{n\, {\rightarrow}\, \infty}\lambda _n^{(q)}=0$. We show that the sequence$\{\lambda _n^{(n-q+1)}\}_{n=q}^{\infty}$converges if$s_r:=\sum_{i=1}^{\infty}{1\over {x_i^r}}<\infty $and${\rm lim}_{n\, {\rightarrow}\, \infty}\lambda _n^{(n-q+1)}\, {\le}\, s_r$. We show also that ifr> 1, then the sequence$\{\lambda _{ln}^{(tn-q+1)}\}_{n=1}^{\infty}$converges and${\rm lim}_{n\, {\rightarrow}\, \infty}\lambda _{ln}^{(tn-q+1)}=0$, wheretandlare given positive integers such thatt≤l−1.

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

2008 ◽  
Vol 01 (01) ◽  
pp. 77-83 ◽  
Author(s):  
Shaofang Hong ◽  
Shaofang Wang
Keyword(s):  
Positive Integer ◽  
Singular Number ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Closed Set ◽  
Prime Factors ◽  
Singular Numbers

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

2021 ◽  
pp. 1-12
Author(s):  
Guangyan Zhu
Keyword(s):  
Greatest Common Divisor ◽  
Text Entry ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Closed Set ◽  
Closed Sets ◽  
Matrix Formula ◽  
Power Matrix ◽  
Positive Integers

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.

A simplified presentation for finding the LCM and the GCF

1974 ◽  
Vol 21 (5) ◽  
pp. 415-416
Author(s):  
Laurence Sherzer
Keyword(s):  
Common Factor ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Prime Factors ◽  
Positive Integers

Given the prime factors of two positive integers, the least common multiple (LCM) of these two numbers is the product of the union of these prime factors, and the greatest common factor (GCF) is the product of the intersection of these prime factors. If we could just state this fact to our students and be understood, our job of teaching them to find the LCM or the GCF of two numbers would be greatly simplified. Unfortunately, as in most teaching, simple verbal statements do not suffice.

A Variation on the Algorithm for GCD and LCM

1982 ◽  
Vol 30 (3) ◽  
pp. 46
Author(s):  
Verna M. Adams
Keyword(s):  
Prime Numbers ◽  
Prime Divisors ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Prime Factors

An algorithm sometimes presented for finding the least common multiple (LCM) of two numbers uses tbe technique of simultaneously finding the prime factors of the numbers. This technique is shown in figure 1. Both numbers are checked for divisibility by 2, then by 3, by 5, and so on. If the divisor does not divide one of the numbers, the number is written on the next line as shown in steps 4 and 5. This process continues until all numbers to the left and on the bottom are prime numbers, or it can be continued, as shown in figure 1, until the numbers across the bottom are all ones. The least common multiple is the product of all of the prime divisors. Thus, LCM (80, 72) = 24 · 32 · 5.

MORE ON A CERTAIN ARITHMETICAL DETERMINANT

2017 ◽  
Vol 97 (1) ◽  
pp. 15-25 ◽  
Author(s):  
ZONGBING LIN ◽  
SIAO HONG
Keyword(s):  
Linear Algebra ◽  
Acta Math ◽  
Arithmetical Function ◽  
Greatest Common Divisor ◽  
Multiplicative Functions ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Mobius Function ◽  
Dirichlet Convolution ◽  
Positive Integers

Let $n\geq 1$ be an integer and $f$ be an arithmetical function. Let $S=\{x_{1},\ldots ,x_{n}\}$ be a set of $n$ distinct positive integers with the property that $d\in S$ if $x\in S$ and $d|x$. Then $\min (S)=1$. Let $(f(S))=(f(\gcd (x_{i},x_{j})))$ and $(f[S])=(f(\text{lcm}(x_{i},x_{j})))$ denote the $n\times n$ matrices whose $(i,j)$-entries are $f$ evaluated at the greatest common divisor of $x_{i}$ and $x_{j}$ and the least common multiple of $x_{i}$ and $x_{j}$, respectively. In 1875, Smith [‘On the value of a certain arithmetical determinant’, Proc. Lond. Math. Soc. 7 (1875–76), 208–212] showed that $\det (f(S))=\prod _{l=1}^{n}(f\ast \unicode[STIX]{x1D707})(x_{l})$, where $f\ast \unicode[STIX]{x1D707}$ is the Dirichlet convolution of $f$ and the Möbius function $\unicode[STIX]{x1D707}$. Bourque and Ligh [‘Matrices associated with classes of multiplicative functions’, Linear Algebra Appl. 216 (1995), 267–275] computed the determinant $\det (f[S])$ if $f$ is multiplicative and, Hong, Hu and Lin [‘On a certain arithmetical determinant’, Acta Math. Hungar. 150 (2016), 372–382] gave formulae for the determinants $\det (f(S\setminus \{1\}))$ and $\det (f[S\setminus \{1\}])$. In this paper, we evaluate the determinant $\det (f(S\setminus \{x_{t}\}))$ for any integer $t$ with $1\leq t\leq n$ and also the determinant $\det (f[S\setminus \{x_{t}\}])$ if $f$ is multiplicative.

scholarly journals Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions

2014 ◽  
Vol 57 (3) ◽  
pp. 551-561 ◽  
Author(s):  
Daniel M. Kane ◽  
Scott Duke Kominers
Keyword(s):  
Lower Bounds ◽  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Finite Arithmetic ◽  
Positive Integers

AbstractFor relatively prime positive integers u0 and r, we consider the least common multiple Ln := lcm(u0, u1..., un) of the finite arithmetic progression . We derive new lower bounds on Ln that improve upon those obtained previously when either u0 or n is large. When r is prime, our best bound is sharp up to a factor of n + 1 for u0 properly chosen, and is also nearly sharp as n → ∞.

scholarly journals The least common multiple of random sets of positive integers

2014 ◽  
Vol 144 ◽  
pp. 92-104 ◽  
Author(s):  
Javier Cilleruelo ◽  
Juanjo Rué ◽  
Paulius Šarka ◽  
Ana Zumalacárregui
Keyword(s):  
Random Sets ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Positive Integers

scholarly journals Nontrivial upper bounds for the least common multiple of an arithmetic progression

2020 ◽  
pp. 2150138
Author(s):  
Sid Ali Bousla
Keyword(s):  
Prime Number ◽  
Arithmetic Progression ◽  
Arithmetic Function ◽  
Upper Bounds ◽  
Addition Formula ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Finite Arithmetic ◽  
Positive Integers ◽  
Coprime Positive Integers

In this paper, we establish some nontrivial and effective upper bounds for the least common multiple of consecutive terms of a finite arithmetic progression. Precisely, we prove that for any two coprime positive integers [Formula: see text] and [Formula: see text], with [Formula: see text], we have [Formula: see text] where [Formula: see text]. If in addition [Formula: see text] is a prime number and [Formula: see text], then we prove that for any [Formula: see text], we have [Formula: see text], where [Formula: see text]. Finally, we apply those inequalities to estimate the arithmetic function [Formula: see text] defined by [Formula: see text] ([Formula: see text]), as well as some values of the generalized Chebyshev function [Formula: see text].

Sharing Teaching Ideas

1988 ◽  
Vol 81 (8) ◽  
pp. 648-652
Author(s):  
Paul A. Klein
Keyword(s):  
Common Factor ◽  
Common Denominator ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Prime Factors ◽  
Instructional Sequences ◽  
Teaching Ideas

Few arithmetic skills have eluded as many students as those used in reducing fractions or in obtaining the least common denominator of two fractions. In the more popular instructional sequences the general concepts of greatest common factor (GCF) and least common multiple (LCM) are developed first Usually, these concepts involve the decomposition of the numbers into prime factors as shown in example 1.

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