scholarly journals On Rainbow Arithmetic Progressions

10.37236/1754 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Maria Axenovich ◽  
Dmitri Fon-Der-Flaass
Keyword(s):  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Natural Numbers

Consider natural numbers $\{1, \cdots, n\}$ colored in three colors. We prove that if each color appears on at least $(n+4)/6$ numbers then there is a three-term arithmetic progression whose elements are colored in distinct colors. This variation on the theme of Van der Waerden's theorem proves the conjecture of Jungić et al.

scholarly journals ON THE DENSITY OF INTEGERS OF THE FORM (p−1)2−n IN ARITHMETIC PROGRESSIONS

2008 ◽  
Vol 78 (3) ◽  
pp. 431-436 ◽  
Author(s):  
XUE-GONG SUN ◽  
JIN-HUI FANG
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Asymptotic Density ◽  
Natural Numbers ◽  
Positive Proportion ◽  
Covering System

AbstractErdős and Odlyzko proved that odd integers k such that k2n+1 is prime for some positive integer n have a positive lower density. In this paper, we characterize all arithmetic progressions in which natural numbers that can be expressed in the form (p−1)2−n (where p is a prime number) have a positive proportion. We also prove that an arithmetic progression consisting of odd numbers can be obtained from a covering system if and only if those integers in such a progression which can be expressed in the form (p−1)2−n have an asymptotic density of zero.

scholarly journals Improved bounds for arithmetic progressions in product sets

2015 ◽  
Vol 11 (08) ◽  
pp. 2295-2303 ◽  
Author(s):  
Dmitrii Zhelezov
Keyword(s):  
Lower Bound ◽  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Complex Numbers ◽  
Natural Numbers ◽  
Multiplicative Constant ◽  
Product Sets

Let B be a set of natural numbers of size n. We prove that the length of the longest arithmetic progression contained in the product set B.B = {bb′|b, b′ ∈ B} cannot be greater than O(n log n) which matches the lower bound provided in an earlier paper up to a multiplicative constant. For sets of complex numbers, we improve the bound to Oϵ(n1 + ϵ) for arbitrary ϵ > 0 assuming the GRH.

Rainbow Arithmetic Progressions and Anti-Ramsey Results

2003 ◽  
Vol 12 (5-6) ◽  
pp. 599-620 ◽  
Author(s):  
V Jungic ◽  
J Licht ◽  
M Mahdian ◽  
J Nesetril ◽  
R Radoicic
Keyword(s):  
Arithmetic Progression ◽  
Ramsey Theory ◽  
Arithmetic Progressions ◽  
Natural Numbers ◽  
Colour Class ◽  
Ramsey Type ◽  
General Perspective

The van der Waerden theorem in Ramsey theory states that, for every k and t and sufficiently large N, every k-colouring of [N] contains a monochromatic arithmetic progression of length t. Motivated by this result, Radoičić conjectured that every equinumerous 3-colouring of [3n] contains a 3-term rainbow arithmetic progression, i.e., an arithmetic progression whose terms are coloured with distinct colours. In this paper, we prove that every 3-colouring of the set of natural numbers for which each colour class has density more than 1/6, contains a 3-term rainbow arithmetic progression. We also prove similar results for colourings of . Finally, we give a general perspective on other anti-Ramsey-type problems that can be considered.

scholarly journals A New Method to Construct Lower Bounds for Van der Waerden Numbers

10.37236/925 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
P. R. Herwig ◽  
M.J.H. Heule ◽  
P. M. Van Lambalgen ◽  
H. Van Maaren
Keyword(s):  
Lower Bounds ◽  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
New Method ◽  
Natural Numbers

We present the Cyclic Zipper Method, a procedure to construct lower bounds for Van der Waerden numbers. Using this method we improved seven lower bounds. For natural numbers $r$, $k$ and $n$ a Van der Waerden certificate $W(r,k,n)$ is a partition of $\{1, \ldots, n\}$ into $r$ subsets, such that none of them contains an arithmetic progression of length $k$ (or larger). Van der Waerden showed that given $r$ and $k$, a smallest $n$ exists - the Van der Waerden number $W(r,k)$ - for which no certificate $W(r,k,n)$ exists. In this paper we investigate Van der Waerden certificates which have certain symmetrical and repetitive properties. Surprisingly, it shows that many Van der Waerden certificates, which must avoid repetitions in terms of arithmetic progressions, reveal strong regularities with respect to several other criteria. The Cyclic Zipper Method exploits these regularities. To illustrate these regularities, two techniques are introduced to visualize certificates.

scholarly journals The existence of small prime gaps in subsets of the integers

2015 ◽  
Vol 11 (03) ◽  
pp. 801-833 ◽  
Author(s):  
Jacques Benatar
Keyword(s):  
Arithmetic Progression ◽  
Prime Numbers ◽  
Arithmetic Progressions ◽  
Natural Numbers ◽  
Positive Proportion ◽  
Small Constant ◽  
Prime Gaps

We consider the problem of finding small prime gaps in various sets [Formula: see text]. Following the work of Goldston–Pintz–Yıldırım, we will consider collections of natural numbers that are well-controlled in arithmetic progressions. Letting qn denote the nth prime in [Formula: see text], we will establish that for any small constant ϵ > 0, the set {qn | qn+1 - qn ≤ ϵ log n} constitutes a positive proportion of all prime numbers. Using the techniques developed by Maynard and Tao, we will also demonstrate that [Formula: see text] has bounded prime gaps. Specific examples, such as the case where [Formula: see text] is an arithmetic progression have already been studied and so the purpose of this paper is to present results for general classes of sets.

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 Obtaining Easily Sums of Powers on Arithmetic Progressions and Properties of Bernoulli Polynomials by Operator Calculus

2017 ◽  
Vol 9 (5) ◽  
pp. 73
Author(s):  
Do Tan Si
Keyword(s):  
Differential Operator ◽  
Generating Function ◽  
Arithmetic Progression ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Arithmetic Progressions ◽  
Operator Calculus ◽  
Sums Of Powers ◽  
The Way

We show that a sum of powers on an arithmetic progression is the transform of a monomial by a differential operator and that its generating function is simply related to that of the Bernoulli polynomials from which consequently it may be calculated. Besides, we show that it is obtainable also from the sums of powers of integers, i.e. from the Bernoulli numbers which in turn may be calculated by a simple algorithm.By the way, for didactic purpose, operator calculus is utilized for proving in a concise manner the main properties of the Bernoulli polynomials. 

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.

scholarly journals The Faulhaber Problem on Sums of Powers on Arithmetic Progressions Resolved

2018 ◽  
Vol 10 (2) ◽  
pp. 5
Author(s):  
Do Tan Si
Keyword(s):  
Arithmetic Progression ◽  
Bernoulli Polynomials ◽  
Arithmetic Progressions ◽  
Power Sums ◽  
Sums Of Powers

We prove that all the Faulhaber coefficients of a sum of odd power of elements of an arithmetic progression may simply be calculated from only one of them which is easily calculable from two Bernoulli polynomials as so as from power sums of integers. This gives two simple formulae for calculating them. As for sums related to even powers, they may be calculated simply from those related to the nearest odd one’s.

scholarly journals NONZERO VALUES OF DIRICHLET L-FUNCTIONS IN VERTICAL ARITHMETIC PROGRESSIONS

2013 ◽  
Vol 09 (04) ◽  
pp. 813-843 ◽  
Author(s):  
GREG MARTIN ◽  
NATHAN NG
Keyword(s):  
Arithmetic Progression ◽  
Positive Constant ◽  
Upper Bound ◽  
Arithmetic Progressions ◽  
Linearly Independent ◽  
Theoretical Evidence

Let L(s, χ) be a fixed Dirichlet L-function. Given a vertical arithmetic progression of T points on the line ℜs = ½, we show that at least cT/ log T of them are not zeros of L(s, χ) (for some positive constant c). This result provides some theoretical evidence towards the conjecture that all nonnegative ordinates of zeros of Dirichlet L-functions are linearly independent over the rationals. We also establish an upper bound (depending upon the progression) for the first member of the arithmetic progression that is not a zero of L(s, χ).

Sign in / Sign up

Export Citation Format

Share Document