scholarly journals On the General Erdős-Turán Conjecture

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Georges Grekos ◽  
Labib Haddad ◽  
Charles Helou ◽  
Jukka Pihko
Keyword(s):  
Finite Sequence ◽  
General Term ◽  
Natural Numbers ◽  
Intrinsic Interest ◽  
Positive Integers ◽  
Increasing Function

The general Erdős-Turán conjecture states that if A is an infinite, strictly increasing sequence of natural numbers whose general term satisfies an≤cn2, for some constant c>0 and for all n, then the number of representations functions of A is unbounded. Here, we introduce the function ψ(n), giving the minimum of the maximal number of representations of a finite sequence A={ak:1≤k≤n} of n natural numbers satisfying ak≤k2 for all k. We show that ψ(n) is an increasing function of n and that the general Erdős-Turán conjecture is equivalent to limn→∞ψ(n)=∞. We also compute some values of ψ(n). We further introduce and study the notion of capacity, which is related to the ψ function by the fact that limn→∞ψ(n) is the capacity of the set of squares of positive integers, but which is also of intrinsic interest.

Finite axiomatizability for equational theories of computable groupoids

1989 ◽  
Vol 54 (3) ◽  
pp. 1018-1022 ◽  
Author(s):  
Peter Perkins
Keyword(s):  
Binary Operation ◽  
Finite Algebra ◽  
Equational Theory ◽  
Natural Numbers ◽  
Finite Axiomatizability ◽  
Finite Algebras ◽  
Equational Theories ◽  
Primitive Recursive ◽  
Positive Integers ◽  
Recursive Set

A computable groupoid is an algebra ‹N, g› where N is the set of natural numbers and g is a recursive (total) binary operation on N. A set L of natural numbers is a computable list of computable groupoids iff L is recursive, ‹N, ϕe› is a computable groupoid for each e ∈ L, and e ∈ L whenever e codes a primitive recursive description of a binary operation on N.Theorem 1. Let L be any computable list of computable groupoids. The set {e ∈ L: the equational theory of ‹N, ϕe› is finitely axiomatizable} is not recursive.Theorem 2. Let S be any recursive set of positive integers. A computable groupoid GS can be constructed so that S is inifinite iff GS has a finitely axiomatizable equational theory.The problem of deciding which finite algebras have finitely axiomatizable equational theories has remained open since it was first posed by Tarski in the early 1960's. Indeed, it is still not known whether the set of such finite algebras is recursively (or corecursively) enumerable. McKenzie [7] has shown that this question of finite axiomatizability for any (finite) algebra of finite type can be reduced to that for a (finite) groupoid.

scholarly journals Partitions of the Natural Numbers

1964 ◽  
Vol 7 (2) ◽  
pp. 219-236 ◽  
Author(s):  
Myer Angel
Keyword(s):  
Natural Numbers ◽  
Positive Integers

We obtain in this article some results concerning partitions of the natural numbers, the most important of which is a generalization of that quoted immediately below. Some intuitive material is included.In 1954, J. Lambek and L. Moser [l] showed that "Two non-decreasing sequences f and g (of non-negative integers) are inverses if and only if the corresponding sets F and G of positive integers, defined by F(m) = the mth element of F = f(m) + m and G(n) = g(n) + n are complementary."

Note on an idea of Fitch

1949 ◽  
Vol 14 (3) ◽  
pp. 175-176 ◽  
Author(s):  
John R. Myhill
Keyword(s):  
Finite Sequence ◽  
Basic Logic ◽  
Natural Numbers ◽  
Recursive Predicate

The concept of a recursively definite predicate of natural numbers was introduced by F. B. Fitch in his An extension of basic logic as follows:Every recursive predicate is recursively definite. If R(x1, …, xn) is recursively definite so is (Ey)R(x1, …, xn−1, y) and (y)R(x1, …, xn−1, y). If R is recursively definite and S is the proper ancestral of R, then S is recursively definite, where the proper ancestral of a relation is defined as follows: if R is of even degree, say 2m, then the proper ancestral of R is the relation S such that for all x1, …, xm, y1, …, ym, S(x1, …, xm, y1, …, ym) is true if and only if there is a finite sequence of sequences (z11, …, zm1), (z12, …, Zm2), …, (z1k, …, Zmk) such that R(Z11, …, Zm1, z12, …, zm2), R(z12, …, zm2, z13, …, zm3), …, R(z1,k−1, z1k, …, Zmk) are all true, where (z11, …, zm1) is (x1, …, xm) and (z1k, …, zmk) is (y1, …, ym).An arithmetic predicate is one which is definable in terms of the operations ‘+’ and ‘·’ of elementary arithmetic, the connectives of the classical prepositional calculus, and quantifiers.

scholarly journals On Waring’s problem: Three cubes and a minicube

2010 ◽  
Vol 200 ◽  
pp. 59-91 ◽  
Author(s):  
Jörg Brüdern ◽  
Trevor D. Wooley
Keyword(s):  
Exponential Sums ◽  
Independent Interest ◽  
Natural Numbers ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Almost All ◽  
Positive Integers

AbstractWe establish that almost all natural numbers n are the sum of four cubes of positive integers, one of which is no larger than n5/36. The proof makes use of an estimate for a certain eighth moment of cubic exponential sums, restricted to minor arcs only, of independent interest.

scholarly journals COUNTING THE NUMBER OF SOLUTIONS TO THE ERDŐS–STRAUS EQUATION ON UNIT FRACTIONS

2013 ◽  
Vol 94 (1) ◽  
pp. 50-105 ◽  
Author(s):  
CHRISTIAN ELSHOLTZ ◽  
TERENCE TAO
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Diophantine Equation ◽  
Upper And Lower Bounds ◽  
Natural Numbers ◽  
Number Of Solutions ◽  
Waring’S Problem ◽  
Unit Fractions ◽  
Image Position ◽  
Positive Integers

AbstractFor any positive integer $n$, let $f(n)$ denote the number of solutions to the Diophantine equation $$\begin{eqnarray*}\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\end{eqnarray*}$$ with $x, y, z$ positive integers. The Erdős–Straus conjecture asserts that $f(n)\gt 0$ for every $n\geq 2$. In this paper we obtain a number of upper and lower bounds for $f(n)$ or $f(p)$ for typical values of natural numbers $n$ and primes $p$. For instance, we establish that $$\begin{eqnarray*}N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\ll \displaystyle \sum _{p\leq N}f(p)\ll N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\log \log N.\end{eqnarray*}$$ These upper and lower bounds show that a typical prime has a small number of solutions to the Erdős–Straus Diophantine equation; small, when compared with other additive problems, like Waring’s problem.

scholarly journals The notion of Natural Numbers among Germanic Mathematicians during the Second Half of the 18th Century

2020 ◽  
Vol 19 (37) ◽  
pp. 01-23
Author(s):  
Elías Fuentes Guillén
Keyword(s):  
19Th Century ◽  
18Th Century ◽  
Number System ◽  
Natural Numbers ◽  
The 19Th Century ◽  
Positive Integers

While the idea of the naturalness of the positive integers is ancient, the idea of the naturals as the foundation of our number system is not. This latter idea, along with other factors, eventually led to the abstract definitions of natural numbers at the end of the 19th century. But, what led to such an idea that was already present among Germanic mathematicians in the first third of the 19th century? This article examines the tensions around the notion of number among the Germanic mathematicians of the second half of the 18th century with the aim of contributing to a better understanding of some of the factors that explain theemergence of such a different approach to naturals.

On Arithmetic Convolutions

1966 ◽  
Vol 9 (3) ◽  
pp. 287-296 ◽  
Author(s):  
T.M. K. Davison
Keyword(s):  
Complex Numbers ◽  
Natural Numbers ◽  
Image Position ◽  
Dirichlet Product ◽  
Positive Integers

Let A be the set of all functions from N, the natural numbers, to C the field of complex numbers. The Dirichlet product of elements f, g of A is given bywhere the summation condition means sum over all positive integers d which divide n.

scholarly journals Record statistics in integer compositions

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Arnold Knopfmacher ◽  
Toufik Mansour
Keyword(s):  
Generating Functions ◽  
Natural Numbers ◽  
Mean Values ◽  
Fixed Subset ◽  
Record Statistics ◽  
Weak Records ◽  
International Audience ◽  
Positive Integers

International audience A $\textit{composition}$ $\sigma =a_1 a_2 \ldots a_m$ of $n$ is an ordered collection of positive integers whose sum is $n$. An element $a_i$ in $\sigma$ is a strong (weak) $\textit{record}$ if $a_i> a_j (a_i \geq a_j)$ for all $j=1,2,\ldots,i-1$. Furthermore, the position of this record is $i$. We derive generating functions for the total number of strong (weak) records in all compositions of $n$, as well as for the sum of the positions of the records in all compositions of $n$, where the parts $a_i$ belong to a fixed subset $A$ of the natural numbers. In particular when $A=\mathbb{N}$, we find the asymptotic mean values for the number, and for the sum of positions, of records in compositions of $n$.

scholarly journals Ramanujan Summation for Powers of Triangular and Pronic Numbers

2020 ◽  
Vol 5 (1-2) ◽  
pp. 05-08
Author(s):  
Dr. R. Sivaraman
Keyword(s):  
Divergent Series ◽  
Natural Numbers ◽  
Real Numbers ◽  
Triangular Numbers ◽  
Positive Integers

The numbers which are sum of first n natural numbers are called Triangular numbers and numbers which are product of two consecutive positive integers are called Pronic numbers. The concept of Ramanujan summation has been dealt by Srinivasa Ramanujan for divergent series of real numbers. In this paper, I will determine the Ramanujan summation for positive integral powers of triangular and Pronic numbers and derive a new compact formula for general case.

Sign in / Sign up

Export Citation Format

Share Document