Additive Number Theory via Approximation by Regular Languages

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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Additive Number Theory via Automata Theory

Theory of Computing Systems ◽

10.1007/s00224-019-09929-9 ◽

2019 ◽

Vol 64 (3) ◽

pp. 542-567 ◽

Cited By ~ 1

Author(s):

Aayush Rajasekaran ◽

Jeffrey Shallit ◽

Tim Smith

Keyword(s):

Number Theory ◽

Automata Theory ◽

Additive Number Theory ◽

Additive Number

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A problem in additive number theory

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100064586 ◽

1988 ◽

Vol 103 (1) ◽

pp. 27-33 ◽

Cited By ~ 28

Author(s):

Jörg Brüdern

Keyword(s):

Number Theory ◽

Additive Number Theory ◽

Interesting Problem ◽

Natural Numbers ◽

Additive Theory ◽

Additive Number ◽

Image Position ◽

Theory Of Numbers

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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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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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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