On the Set of Common Differences in van der Waerden’s Theorem on Arithmetic Progressions

1999 ◽  
Vol 42 (1) ◽  
pp. 25-36 ◽  
Author(s):  
Tom C. Brown ◽  
Ronald L. Graham ◽  
Bruce M. Landman
Keyword(s):  
Positive Integer ◽  
Arithmetic Progression ◽  
Infinite Number ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Arithmetic Progressions ◽  
Large Set ◽  
Open Questions ◽  
The Family ◽  
Positive Integers

AbstractAnalogues of van derWaerden’s theorem on arithmetic progressions are considered where the family of all arithmetic progressions, AP, is replaced by some subfamily of AP. Specifically, we want to know for which sets A, of positive integers, the following statement holds: for all positive integers r and k, there exists a positive integer n = w′(k, r) such that for every r-coloring of [1, n] there exists a monochromatic k-term arithmetic progression whose common difference belongs to A. We will call any subset of the positive integers that has the above property large. A set having this property for a specific fixed r will be called r-large. We give some necessary conditions for a set to be large, including the fact that every large set must contain an infinite number of multiples of each positive integer. Also, no large set {an : n = 1, 2,…} can have . Sufficient conditions for a set to be large are also given. We show that any set containing n-cubes for arbitrarily large n, is a large set. Results involving the connection between the notions of “large” and “2-large” are given. Several open questions and a conjecture are presented.

On Sums of Progressions of Positive Integers

1970 ◽  
Vol 54 (388) ◽  
pp. 113-115
Author(s):  
R. L. Goodstein
Keyword(s):  
Positive Integer ◽  
Arithmetic Progression ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Common ◽  
Necessary And Sufficient ◽  
Positive Integers

We consider the problem of finding necessary and sufficient conditions for a positive integer to be the sum of an arithmetic progression of positive integers with a given common difference, starting with the case when the common difference is unity.

scholarly journals Minimum Number of Colours to Avoid k-Term Monochromatic Arithmetic Progressions

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 247
Author(s):  
Kai An Sim ◽  
Kok Bin Wong
Keyword(s):  
Positive Integer ◽  
Arithmetic Progression ◽  
Upper Bound ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Arithmetic Progressions ◽  
Minimum Number ◽  
Necessary And Sufficient

By recalling van der Waerden theorem, there exists a least a positive integer w=w(k;r) such that for any n≥w, every r-colouring of [1,n] admits a monochromatic k-term arithmetic progression. Let k≥2 and rk(n) denote the minimum number of colour required so that there exists a rk(n)-colouring of [1,n] that avoids any monochromatic k-term arithmetic progression. In this paper, we give necessary and sufficient conditions for rk(n+1)=rk(n). We also show that rk(n)=2 for all k≤n≤2(k−1)2 and give an upper bound for rp(pm) for any prime p≥3 and integer m≥2.

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 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 Characterizations and Identities for Isosceles Triangular Numbers

2021 ◽  
Vol 14 (2) ◽  
pp. 380-395
Author(s):  
Jiramate Punpim ◽  
Somphong Jitman
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Triangular Numbers ◽  
Triangular Number ◽  
Recursive Formulas ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Special Case

Triangular numbers have been of interest and continuously studied due to their beautiful representations, nice properties, and various links with other figurate numbers. For positive integers n and l, the nth l-isosceles triangular number is a generalization of triangular numbers defined to be the arithmetic sum of the formT(n, l) = 1 + (1 + l) + (1 + 2l) + · · · + (1 + (n − 1)l).In this paper, we focus on characterizations and identities for isosceles triangular numbers as well as their links with other figurate numbers. Recursive formulas for constructions of isosceles triangular numbers are given together with necessary and sufficient conditions for a positive integer to be a sum of isosceles triangular  numbers. Various identities for isosceles triangular numbers are established. Results on triangular numbers can be viewed as a special case.

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 Integral quadratic forms avoiding arithmetic progressions

2020 ◽  
Vol 16 (10) ◽  
pp. 2141-2148
Author(s):  
A. G. Earnest ◽  
Ji Young Kim
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Arithmetic Progressions ◽  
Integral Quadratic Form ◽  
Positive Integers

For every positive integer [Formula: see text], it is shown that there exists a positive definite diagonal quaternary integral quadratic form that represents all positive integers except for precisely those which lie in [Formula: see text] arithmetic progressions. For [Formula: see text], all forms with this property are determined.

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 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 On powerful numbers

1986 ◽  
Vol 9 (4) ◽  
pp. 801-806 ◽  
Author(s):  
R. A. Mollin ◽  
P. G. Walsh
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Effective Algorithm ◽  
Existence Proof ◽  
Prime Decomposition ◽  
Open Questions ◽  
Necessary And Sufficient ◽  
Powerful Numbers

A powerful number is a positive integernsatisfying the property thatp2dividesnwhenever the primepdividesn; i.e., in the canonical prime decomposition ofn, no prime appears with exponent 1. In [1], S.W. Golomb introduced and studied such numbers. In particular, he asked whether(25,27)is the only pair of consecutive odd powerful numbers. This question was settled in [2] by W.A. Sentance who gave necessary and sufficient conditions for the existence of such pairs. The first result of this paper is to provide a generalization of Sentance's result by giving necessary and sufficient conditions for the existence of pairs of powerful numbers spaced evenly apart. This result leads us naturally to consider integers which are representable as a proper difference of two powerful numbers, i.e.n=p1−p2wherep1andp2are powerful numbers with g.c.d.(p1,p2)=1. Golomb (op.cit.) conjectured that6is not a proper difference of two powerful numbers, and that there are infinitely many numbers which cannot be represented as a proper difference of two powerful numbers. The antithesis of this conjecture was proved by W.L. McDaniel [3] who verified that every non-zero integer is in fact a proper difference of two powerful numbers in infinitely many ways. McDaniel's proof is essentially an existence proof. The second result of this paper is a simpler proof of McDaniel's result as well as an effective algorithm (in the proof) for explicitly determining infinitely many such representations. However, in both our proof and McDaniel's proof one of the powerful numbers is almost always a perfect square (namely one is always a perfect square whenn≢2(mod4)). We provide in §2 a proof that all even integers are representable in infinitely many ways as a proper nonsquare difference; i.e., proper difference of two powerful numbers neither of which is a perfect square. This, in conjunction with the odd case in [4], shows that every integer is representable in infinitely many ways as a proper nonsquare difference. Moreover, in §2 we present some miscellaneous results and conclude with a discussion of some open questions.

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