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 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].

scholarly journals The least common multiple of consecutive arithmetic progression terms

2011 ◽  
Vol 54 (2) ◽  
pp. 431-441 ◽  
Author(s):  
Shaofang Hong ◽  
Guoyou Qian
Keyword(s):  
Positive Integer ◽  
Periodic Function ◽  
Arithmetic Progression ◽  
Arithmetic Function ◽  
Arithmetic Progressions ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Image Position ◽  
Positive Integers

AbstractLet k ≥ 0, a ≥ 1 and b ≥ 0 be integers. We define the arithmetic function gk,a,b for any positive integer n byIf we let a = 1 and b = 0, then gk,a,b becomes the arithmetic function that was previously introduced by Farhi. Farhi proved that gk,1,0 is periodic and that k! is a period. Hong and Yang improved Farhi's period k! to lcm(1, 2, … , k) and conjectured that (lcm(1, 2, … , k, k + 1))/(k + 1) divides the smallest period of gk,1,0. Recently, Farhi and Kane proved this conjecture and determined the smallest period of gk,1,0. For the general integers a ≥ 1 and b ≥ 0, it is natural to ask the following interesting question: is gk,a,b periodic? If so, what is the smallest period of gk,a,b? We first show that the arithmetic function gk,a,b is periodic. Subsequently, we provide detailed p-adic analysis of the periodic function gk,a,b. Finally, we determine the smallest period of gk,a,b. Our result extends the Farhi–Kane Theorem from the set of positive integers to general arithmetic progressions.

scholarly journals The Minimal Number of Three-Term Arithmetic Progressions Modulo a Prime Converges to a Limit

2008 ◽  
Vol 51 (1) ◽  
pp. 47-56 ◽  
Author(s):  
Ernie Croot
Keyword(s):  
Lower Bounds ◽  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Famous Conjecture ◽  
Positive Integers

AbstractHow few three-term arithmetic progressions can a subset S ⊆ ℤN := ℤ/Nℤ have if |S| ≥ υN (that is, S has density at least υ)? Varnavides showed that this number of arithmetic progressions is at least c(υ)N2 for sufficiently large integers N. It is well known that determining good lower bounds for c(υ) > 0 is at the same level of depth as Erdös's famous conjecture about whether a subset T of the naturals where Σn∈T 1/n diverges, has a k-term arithmetic progression for k = 3 (that is, a three-term arithmetic progression).We answer a question posed by B. Green about how this minimial number of progressions oscillates for a fixed density υ as N runs through the primes, and as N runs through the odd positive integers.

scholarly journals SUMS OF PRODUCTS OF CONGRUENCE CLASSES AND OF ARITHMETIC PROGRESSIONS

2009 ◽  
Vol 05 (04) ◽  
pp. 625-634
Author(s):  
SERGEI V. KONYAGIN ◽  
MELVYN B. NATHANSON
Keyword(s):  
Arithmetic Progression ◽  
Congruence Class ◽  
Arithmetic Progressions ◽  
Sum Of Products ◽  
Sums Of Products ◽  
Positive Integers ◽  
Congruence Classes

Consider the congruence class Rm(a) = {a + im : i ∈ Z} and the infinite arithmetic progression Pm(a) = {a + im : i ∈ N0}. For positive integers a,b,c,d,m the sum of products set Rm(a)Rm(b) + Rm(c)Rm(d) consists of all integers of the form (a+im) · (b+jm)+(c+km)(d+ℓm) for some i,j,k,ℓ ∈ Z. It is proved that if gcd (a,b,c,d,m) = 1, then Rm(a)Rm(b) + Rm(c)Rm(d) is equal to the congruence class Rm(ab+cd), and that the sum of products set Pm(a)Pm(b)+Pm(c)Pm eventually coincides with the infinite arithmetic progression Pm(ab+cd).

scholarly journals Monochromatic arithmetic progressions with large differences

1999 ◽  
Vol 60 (1) ◽  
pp. 21-35
Author(s):  
Tom C. Brown ◽  
Bruce M. Landman
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Arithmetic Progression ◽  
Constant Function ◽  
Arithmetic Progressions ◽  
Upper And Lower Bounds

A generalisation of the van der Waerden numbers w(k, r) is considered. For a function f: Z+ → R+ define w(f, k, r) to be the least positive integer (if it exists) such that for every r-coloring of [1, w(f, k, r)] there is a monochromatic arithmetic progression {a + id: 0 ≤ i ≤ k −1} such that d ≥ f(a). Upper and lower bounds are given for w(f, 3, 2). For k > 3 or r > 2, particular functions f are given such that w(f, k, r) does not exist. More results are obtained for the case in which f is a constant function.

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 Sets of Integers that do not Contain Long Arithmetic Progressions

10.37236/546 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Kevin O'Bryant
Keyword(s):  
Lower Bounds ◽  
Arithmetic Progression ◽  
Arithmetic Progressions

Combining ideas of Rankin, Elkin, Green & Wolf, we give constructive lower bounds for $r_k(N)$, the largest size of a subset of $\{1,2,\dots,N\}$ that does not contain a $k$-element arithmetic progression: For every $\epsilon>0$, if $N$ is sufficiently large, then $$r_3(N) \geq N \left(\frac{6\cdot 2^{3/4} \sqrt{5}}{e \,\pi^{3/2}}-\epsilon\right) \exp\left({-\sqrt{8\log N}+\tfrac14\log\log N}\right),$$ $$r_k(N) \geq N \, C_k\,\exp\left({-n 2^{(n-1)/2} \sqrt[n]{\log N}+\tfrac{1}{2n}\log\log N}\right),$$ where $C_k>0$ is an unspecified constant, $\log=\log_2$, $\exp(x)=2^x$, and $n=\lceil{\log k}\rceil$. These are currently the best lower bounds for all $k$, and are an improvement over previous lower bounds for all $k\neq4$.

scholarly journals On the least common multiple of polynomial sequences at prime arguments

2021 ◽  
pp. 1-11
Author(s):  
Ayan Nath ◽  
Abhishek Jha
Keyword(s):  
Lower Bounds ◽  
Prime Divisor ◽  
Irreducible Polynomial ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Polynomial Sequences

Cilleruelo conjectured that if [Formula: see text] is an irreducible polynomial of degree [Formula: see text] then, [Formula: see text] In this paper, we investigate the analog of prime arguments, namely, [Formula: see text] where [Formula: see text] denotes a prime and obtain nontrivial lower bounds on it. Further, we also show some results regarding the greatest prime divisor of [Formula: see text]

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