scholarly journals Analytic Erdös-Turán conjectures and Erdös-Fuchs theorem

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.

A problem in additive number theory

1988 ◽  
Vol 103 (1) ◽  
pp. 27-33 ◽  
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.

scholarly journals Computers as Fresh Mathematical Reality. II. Waring Problem

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.

scholarly journals Hypertranscendental elements of a formal power-series ring of positive characteristic

1992 ◽  
Vol 125 ◽  
pp. 93-103 ◽  
Author(s):  
Kayoko Shikishima-Tsuji ◽  
Masashi Katsura
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Positive Characteristic ◽  
Rational Numbers ◽  
Natural Numbers ◽  
Real Numbers ◽  
Power Series Ring ◽  
Series Ring ◽  
Formal Power Series Ring ◽  
Formal Power

Throughout this paper, we denote by N, Q and R the set of all natural numbers containing 0, the set of all rational numbers, and the set of all real numbers, respectively.

scholarly journals Noncommutative rational Pólya series

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Jason Bell ◽  
Daniel Smertnig
Keyword(s):  
Number Theory ◽  
Power Series ◽  
Formal Power Series ◽  
Finite Automaton ◽  
Automata Theory ◽  
Finitely Generated ◽  
Noncommutative Algebra ◽  
Rational Series ◽  
Unit Equations ◽  
Formal Power

AbstractA (noncommutative) Pólya series over a field K is a formal power series whose nonzero coefficients are contained in a finitely generated subgroup of $$K^\times $$ K × . We show that rational Pólya series are unambiguous rational series, proving a 40 year old conjecture of Reutenauer. The proof combines methods from noncommutative algebra, automata theory, and number theory (specifically, unit equations). As a corollary, a rational series is a Pólya series if and only if it is Hadamard sub-invertible. Phrased differently, we show that every weighted finite automaton taking values in a finitely generated subgroup of a field (and zero) is equivalent to an unambiguous weighted finite automaton.

scholarly journals Additive Number Theory via Approximation by Regular Languages

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.

scholarly journals On transformations of formal power series

2003 ◽  
Vol 184 (2) ◽  
pp. 369-383 ◽  
Author(s):  
Manfred Droste ◽  
Guo-Qiang Zhang
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Formal Power

scholarly journals On the Square Root of a Bell Matrix

2021 ◽  
Vol 76 (1) ◽  
Author(s):  
Donatella Merlini
Keyword(s):  
Power Series ◽  
Closed Form ◽  
Formal Power Series ◽  
Computational Method ◽  
Square Root ◽  
Riordan Arrays ◽  
Formal Power

AbstractIn the context of Riordan arrays, the problem of determining the square root of a Bell matrix $$R={\mathcal {R}}(f(t)/t,\ f(t))$$ R = R ( f ( t ) / t , f ( t ) ) defined by a formal power series $$f(t)=\sum _{k \ge 0}f_kt^k$$ f ( t ) = ∑ k ≥ 0 f k t k with $$f(0)=f_0=0$$ f ( 0 ) = f 0 = 0 is presented. It is proved that if $$f^\prime (0)=1$$ f ′ ( 0 ) = 1 and $$f^{\prime \prime }(0)\ne 0$$ f ″ ( 0 ) ≠ 0 then there exists another Bell matrix $$H={\mathcal {R}}(h(t)/t,\ h(t))$$ H = R ( h ( t ) / t , h ( t ) ) such that $$H*H=R;$$ H ∗ H = R ; in particular, function h(t) is univocally determined by a symbolic computational method which in many situations allows to find the function in closed form. Moreover, it is shown that function h(t) is related to the solution of Schröder’s equation. We also compute a Riordan involution related to this kind of matrices.

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