A problem in additive number theory

The determination of the minimal s such that all large natural numbers n admit a representation asis an interesting problem in the additive theory of numbers and has a considerable literature, For historical comments the reader is referred to the author's paper [2] where the best currently known result is proved. The purpose here is a further improvement.

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On Waring's problem for cubes

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100069711 ◽

1991 ◽

Vol 109 (2) ◽

pp. 229-256 ◽

Cited By ~ 10

Author(s):

Jörg Brüdern

Keyword(s):

Singular Series ◽

Natural Numbers ◽

Additive Theory ◽

Waring’S Problem ◽

Waring's Problem ◽

Quantitative Form ◽

Image Position ◽

Theory Of Numbers

A classical conjecture in the additive theory of numbers is that all sufficiently large natural numbers may be written as the sum of four positive cubes of integers. This is known as the Four Cubes Problem, and since the pioneering work of Hardy and Littlewood one expects a much more precise quantitative form of the conjecture to hold. Let v(n) be the number of representations of n in the proposed manner. Then the expected formula takes the shapewhere (n) is the singular series associated with four cubes as familiar in the Hardy–Littlewood theory.

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Computers as Fresh Mathematical Reality. II. Waring Problem

Computer Tools in Education ◽

10.32603/2071-2340-2020-3-5-55 ◽

2020 ◽

pp. 5-55

Author(s):

Nikolai Vavilov ◽

Keyword(s):

Number Theory ◽

Classical Problem ◽

Additive Number Theory ◽

Natural Numbers ◽

Additive Number ◽

Waring Problem ◽

History Of ◽

Almost All ◽

Xix Century

In this part I discuss the role of computers in the current research on the additive number theory, in particular in the solution of the classical Waring problem. In its original XVIII century form this problem consisted in finding for each natural k the smallest such s=g(k) that all natural numbers n can be written as sums of s non-negative k-th powers, n=x_1^k+ldots+x_s^k. In the XIX century the problem was modified as the quest of finding such minimal $s=G(k)$ that almost all n can be expressed in this form. In the XX century this problem was further specified, as for finding such G(k) and the precise list of exceptions. The XIX century problem is still unsolved even or cubes. However, even the solution of the original Waring problem was [almost] finalised only in 1984, with heavy use of computers. In the present paper we document the history of this classical problem itself and its solution, as also discuss possibilities of using this and surrounding material in education, and some further related aspects.

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THE ISOPERIMETRIC METHOD IN NON-ABELIAN GROUPS WITH AN APPLICATION TO OPTIMALLY SMALL SUMSETS

International Journal of Number Theory ◽

10.1142/s1793042108001821 ◽

2008 ◽

Vol 04 (06) ◽

pp. 927-958 ◽

Cited By ~ 1

Author(s):

ÉRIC BALANDRAUD

Keyword(s):

Number Theory ◽

Abelian Group ◽

Abelian Groups ◽

Solvable Groups ◽

Additive Number Theory ◽

Alternating Groups ◽

Additive Number ◽

New Interpretation ◽

Set Addition

Set addition theory is born a few decades ago from additive number theory. Several difficult issues, more combinatorial in nature than algebraic, have been revealed. In particular, computing the values taken by the function: [Formula: see text] where G is a given group does not seem easy in general. Some successive results, using Kneser's Theorem, allowed the determination of the values of this function, provided that the group G is abelian. Recently, a method called isoperimetric, has been developed by Hamidoune and allowed new proofs and generalizations of the classical theorems in additive number theory. For instance, a new interpretation of the isoperimetric method has been able to give a new proof of Kneser's Theorem. The purpose of this article is to adapt this last proof in a non-abelian group, in order to give new values of the function μG, for some solvable groups and alternating groups. These values allow us in particular to answer negatively a question asked in the literature on the μG functions.

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Analytic Erdös-Turán conjectures and Erdös-Fuchs theorem

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/ijmms.2005.3767 ◽

2005 ◽

Vol 2005 (23) ◽

pp. 3767-3780

Author(s):

L. Haddad ◽

C. Helou ◽

J. Pihko

Keyword(s):

Number Theory ◽

Power Series ◽

Formal Power Series ◽

Positive Constant ◽

Additive Number Theory ◽

Natural Numbers ◽

Additive Number ◽

Bounded Below ◽

Formal Power

We consider and study formal power series, that we call supported series, with real coefficients which are either zero or bounded below by some positive constant. The sequences of such coefficients have a lot of similarity with sequences of natural numbers considered in additive number theory. It is this analogy that we pursue, thus establishing many properties and giving equivalent statements to the well-known Erdös-Turán conjectures in terms of supported series and extending to them a version of Erdös-Fuchs theorem.

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Additive Number Theory via Approximation by Regular Languages

International Journal of Foundations of Computer Science ◽

10.1142/s0129054120410014 ◽

2020 ◽

Vol 31 (06) ◽

pp. 667-687

Author(s):

Jason Bell ◽

Thomas F. Lidbetter ◽

Jeffrey Shallit

Keyword(s):

Number Theory ◽

Natural Number ◽

Formal Languages ◽

Regular Languages ◽

Automata Theory ◽

Additive Number Theory ◽

Text Representation ◽

Natural Numbers ◽

Additive Number ◽

Number Formula

We prove some new theorems in additive number theory, using novel techniques from automata theory and formal languages. As an example of our method, we prove that every natural number [Formula: see text] is the sum of at most three natural numbers whose base-[Formula: see text] representation has an equal number of [Formula: see text]’s and [Formula: see text]’s.

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