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 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.

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.

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.

On some characteristics of subset of prime numbers

2019 ◽  
pp. 99-109
Author(s):  
V. Malakhovsky
Keyword(s):  
Arithmetic Progression ◽  
Prime Numbers ◽  
Natural Numbers

The set of prime numbers p ≥ 5 is divided into two nonoverlapping subset P1 = {6k1  1}, P2 = {6k2 + 1}, where ki ⋲ A (i = 1,2). Subsets A1, A2 of natural numbers is defined by differences Ai = N\Bi, where B1, B2 are subset {j1}, {j2} defining subsets {6j1 – 1}, {6j2 + 1} of odd composite numbers. In [1] is proved two theorems permitting easily find by means of arithmetic progression subset Bi for ji  a ⋲ N. The tables of numbers ki for a = 500 are defined and some characteristic of subsets P1, P2 are given.

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 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).

The diophantine equation x2 + paqb = yq

2021 ◽  
pp. 1-18
Author(s):  
Hemar Godinho ◽  
Victor G. L. Neumann
Keyword(s):  
Diophantine Equation ◽  
Prime Numbers ◽  
Natural Numbers ◽  
Lucas Sequences ◽  
Primitive Divisors

In this paper, we consider the Diophantine equation in the title, where [Formula: see text] are distinct odd prime numbers and [Formula: see text] are natural numbers. We present many results given conditions for the existence of integers solutions for this equation, according to the values of [Formula: see text] and [Formula: see text]. Our methods are elementary in nature and are based upon the study of the primitive divisors of certain Lucas sequences as well as the factorization of certain polynomials.

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 III. The sums of the series of the reciprocals of the prime numbers and of their powers

1882 ◽  
Vol 33 (216-219) ◽  
pp. 4-10 ◽  
Keyword(s):  
Prime Numbers ◽  
Natural Numbers ◽  
The Difference

Euler has shown that it is possible to sum the series of reciprocals of powers of the prime numbers, and he has calculated the values of these sums for the even powers. I thought it of some interest to calculate the sums for the odd powers, and to evaluate a peculiar constant (somewhat analogous to the Eulerian constant,— γ = 0·57721 56649 01532 86060 65) which presents itself, in the series of simple reciprocals of primes, as the difference between the sum of the series and the double logarithmic infinity to the Napierian base ϵ. The summation of these series was shown by Euler to depend upon the Napierian logarithms of the sums of the reciprocals of the powers of the natural numbers.

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