scholarly journals Convergence of the Collatz Sequence

2019 ◽  
Vol 17 ◽  
pp. 19-33
Author(s):  
Anatoliy Nikolaychuk
Keyword(s):  
Natural Number ◽  
Bounded Sequence ◽  
New Method ◽  
Numerical Parameter ◽  
Natural Numbers

For any natural number was created the supplement sequence, that is convergent together with the original Collatz sequence. The numerical parameter - index was defined, that is the same for both sequences. This new method provides the following results: All natural numbers were distributed into six different classes; The properties of index were found for the different classes; For any natural number was constructed the bounded sequence of increasing numbers,     that is convergent together with the regular Collatz sequence.

A finite arithmetic

1973 ◽  
Vol 38 (2) ◽  
pp. 232-248 ◽  
Author(s):  
Philip T. Shepard
Keyword(s):  
Number Theory ◽  
Natural Number ◽  
Singular Term ◽  
New Method ◽  
Axiomatic System ◽  
Natural Numbers ◽  
Substantial Loss ◽  
First Order ◽  
Finite Arithmetic

In this paper I shall argue that the presumption of infinitude may be excised from the area of mathematics known as natural number theory with no substantial loss. Except for a few concluding remarks, I shall restrict my concern in here arguing the thesis to the business of constructing and developing a first-order axiomatic system for arithmetic (called ‘FA’ for finite arithmetic) that contains no theorem to the effect that there are infinitely many numbers.The paper will consist of five parts. Part I characterizes the underlying logic of FA. In part II ordering of natural numbers is developed from a restricted set of axioms, induction schemata are proved and a way of expressing finitude presented. A full set of axioms are used in part III to prove working theorems on comparison of size. In part IV an ordinal expression is defined and characteristic theorems proved. Theorems for addition and multiplication are derived in part V from definitions in terms of the ordinal expression of part IV. The crucial final constructions of part V present a new method of replacing recursive characterizations by strict definitions.In view of our resolution not to assume that there are infinitely many numbers, we shall have to deal with the situation where singular arithmetic terms of FA may fail to refer. For I know of no acceptable and systematic way of avoiding such situations. As a further result, singular-term instances of universal generalizations of FA are not to be inferred directly from the generalizations themselves. Nevertheless, (i) (x)(y)(x + y = y + x), for example, and all its instances are provable in FA.

scholarly journals ORDER CONVERGENCE OF ORDER BOUNDED SEQUENCES IN RIESZ SPACES

1998 ◽  
Vol 29 (1) ◽  
pp. 41-45
Author(s):  
BORIS LAVRIC
Keyword(s):  
Natural Number ◽  
Bounded Sequence ◽  
Recursive Relation ◽  
Sufficient Condition ◽  
Order Convergence ◽  
Natural Numbers ◽  
Real Numbers ◽  
Relative Prime ◽  
Order Boundedness ◽  
Bounded Sequences

We consider sequences  in a Dedekind $\sigma$-complete Riese space, satisfying a recursive relation \[ x_{n+p}\ge \sum_{j=1}^p \alpha_{n,j} x_{n+p-j} \qquad \text{for } n=1, 2, \cdots\] where $p$ is a given natural number and $\alpha_{n,j}$ are nonnegative real numbers satisfying $\sum_{j=1}^p\alpha_{n,j}=1$. We obtain a sufficient condition on coefficients $\alpha_{n,j}$ for which order boundedness of such a sequence $(x_n)_{n=1}^\infty$ implies its order convergence. In a particular case when $\alpha_{n,j}=\alpha_{j}$ for all $n$ and $j$, it is shown that every order bounded sequence satisfying the above recursive relation order converges if and only if natural numbers $j \le p$ for which $\alpha_{j}>0$, are relative prime.

scholarly journals Supplement to the Collatz Conjecture

2018 ◽  
Vol 15 ◽  
pp. 8120-8132
Author(s):  
Anatoliy Nikolaychuk
Keyword(s):  
Natural Number ◽  
New Method ◽  
Natural Numbers ◽  
Original Sequence ◽  
Collatz Conjecture

For any natural number was created the supplement sequence, that is convergent together with the original sequence. The parameter - index was defined, that is the same tor both sequences. This new method provides the following results: All natural numbers were distributed into different classes according to the corresponding indexes; The analytic formulas ( not by computer performed routine calculations) were produced, the formulas for groups of consecutive natural numbers of different lengths, having the same index; The new algorithm to find index for any natural number was constructed and proved.

The Natural Numbers

2018 ◽  
Author(s):  
Øystein Linnebo
Keyword(s):  
Natural Number ◽  
Peano Arithmetic ◽  
Number Sequence ◽  
Natural Numbers ◽  
Ordinal Numbers ◽  
Cardinal Numbers

How are the natural numbers individuated? That is, what is our most basic way of singling out a natural number for reference in language or in thought? According to Frege and many of his followers, the natural numbers are cardinal numbers, individuated by the cardinalities of the collections that they number. Another answer regards the natural numbers as ordinal numbers, individuated by their positions in the natural number sequence. Some reasons to favor the second answer are presented. This answer is therefore developed in more detail, involving a form of abstraction on numerals. Based on this answer, a justification for the axioms of Dedekind–Peano arithmetic is developed.

scholarly journals A unified treatment of syntax with binders

2012 ◽  
Vol 22 (4-5) ◽  
pp. 614-704 ◽  
Author(s):  
NICOLAS POUILLARD ◽  
FRANÇOIS POTTIER
Keyword(s):  
Natural Number ◽  
Data Structures ◽  
Transformation Function ◽  
Natural Numbers ◽  
Programming Error ◽  
Free Variables ◽  
De Bruijn ◽  
Abstract Interface ◽  
Representation Techniques ◽  
De Bruijn Indices

AbstractAtoms and de Bruijn indices are two well-known representation techniques for data structures that involve names and binders. However, using either technique, it is all too easy to make a programming error that causes one name to be used where another was intended. We propose an abstract interface to names and binders that rules out many of these errors. This interface is implemented as a library in Agda. It allows defining and manipulating term representations in nominal style and in de Bruijn style. The programmer is not forced to choose between these styles: on the contrary, the library allows using both styles in the same program, if desired. Whereas indexing the types of names and terms with a natural number is a well-known technique to better control the use of de Bruijn indices, we index types with worlds. Worlds are at the same time more precise and more abstract than natural numbers. Via logical relations and parametricity, we are able to demonstrate in what sense our library is safe, and to obtain theorems for free about world-polymorphic functions. For instance, we prove that a world-polymorphic term transformation function must commute with any renaming of the free variables. The proof is entirely carried out in Agda.

scholarly journals Continued Fractions of Square Roots of Natural Numbers

10.14311/1821 ◽  
2013 ◽  
Vol 53 (4) ◽  
Author(s):  
L'ubomíra Balková ◽  
Aranka Hrušková
Keyword(s):  
Natural Number ◽  
Continued Fractions ◽  
Natural Numbers ◽  
Precise Form ◽  
Square Roots

In this paper, we will first summarize known results concerning continued fractions. Then we will limit our consideration to continued fractions of quadratic numbers. The second author describes periods and sometimes the precise form of continued fractions of ?N, where N is a natural number. In cases where we have been able to find such results in the literature, we recall the original authors, however many results seem to be new.

Compact serialization of Prolog terms (with catalan skeletons, cantor tupling and Gödel numberings)

2013 ◽  
Vol 13 (4-5) ◽  
pp. 847-861 ◽  
Author(s):  
PAUL TARAU
Keyword(s):  
Natural Number ◽  
Memory Representation ◽  
Ranking Algorithm ◽  
Natural Numbers ◽  
Algorithm Mapping ◽  
Data Objects ◽  
Novel Algorithm

AbstractWe describe a compact serialization algorithm mapping Prolog terms to natural numbers of bit-sizes proportional to the memory representation of the terms. The algorithm is a ‘no bit lost’ bijection, as it associates to each Prolog term a unique natural number and each natural number corresponds to a unique syntactically well-formed term.To avoid an exponential explosion resulting from bijections mapping term trees to natural numbers, we separate the symbol content and the syntactic skeleton of a term that we serialize compactly using a ranking algorithm for Catalan families.A novel algorithm for the generalized Cantor bijection between ${\mathbb{N}$ and ${\mathbb{N}$k is used in the process of assigning polynomially bounded Gödel numberings to various data objects involved in the translation.

THE EXCEPTIONAL SET IN THE POLYNOMIAL GOLDBACH PROBLEM

2011 ◽  
Vol 07 (03) ◽  
pp. 579-591 ◽  
Author(s):  
PAUL POLLACK
Keyword(s):  
Asymptotic Formula ◽  
Finite Field ◽  
Natural Number ◽  
Riemann Hypothesis ◽  
Natural Numbers ◽  
Exceptional Set ◽  
Goldbach Problem

For each natural number N, let R(N) denote the number of representations of N as a sum of two primes. Hardy and Littlewood proposed a plausible asymptotic formula for R(2N) and showed, under the assumption of the Riemann Hypothesis for Dirichlet L-functions, that the formula holds "on average" in a certain sense. From this they deduced (under ERH) that all but Oϵ(x1/2+ϵ) of the even natural numbers in [1, x] can be written as a sum of two primes. We generalize their results to the setting of polynomials over a finite field. Owing to Weil's Riemann Hypothesis, our results are unconditional.

scholarly journals On Certain Axiomatizations of Arithmetic of Natural and Integer Numbers

Axioms ◽  
2019 ◽  
Vol 8 (3) ◽  
pp. 103 ◽  
Author(s):  
Urszula Wybraniec-Skardowska
Keyword(s):  
Natural Number ◽  
Ordered Sets ◽  
Natural Numbers ◽  
Elementary Theories

The systems of arithmetic discussed in this work are non-elementary theories. In this paper, natural numbers are characterized axiomatically in two different ways. We begin by recalling the classical set P of axioms of Peano’s arithmetic of natural numbers proposed in 1889 (including such primitive notions as: set of natural numbers, zero, successor of natural number) and compare it with the set W of axioms of this arithmetic (including the primitive notions like: set of natural numbers and relation of inequality) proposed by Witold Wilkosz, a Polish logician, philosopher and mathematician, in 1932. The axioms W are those of ordered sets without largest element, in which every non-empty set has a least element, and every set bounded from above has a greatest element. We show that P and W are equivalent and also that the systems of arithmetic based on W or on P, are categorical and consistent. There follows a set of intuitive axioms PI of integers arithmetic, modelled on P and proposed by B. Iwanuś, as well as a set of axioms WI of this arithmetic, modelled on the W axioms, PI and WI being also equivalent, categorical and consistent. We also discuss the problem of independence of sets of axioms, which were dealt with earlier.

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