scholarly journals Maximum Number of Steps Taken by Modular Exponentiation and Euclidean Algorithm

2019 ◽  
Vol 27 (1) ◽  
pp. 87-91
Author(s):  
Hiroyuki Okazaki ◽  
Koh-ichi Nagao ◽  
Yuichi Futa
Keyword(s):  
Computational Complexity ◽  
Natural Number ◽  
Euclidean Algorithm ◽  
Negative Integer ◽  
Modular Exponentiation ◽  
Natural Numbers ◽  
Complexity Of Algorithms ◽  
Modular Products

Summary In this article we formalize in Mizar [1], [2] the maximum number of steps taken by some number theoretical algorithms, “right–to–left binary algorithm” for modular exponentiation and “Euclidean algorithm” [5]. For any natural numbers a, b, n, “right–to–left binary algorithm” can calculate the natural number, see (Def. 2), AlgoBPow(a, n, m) := ab mod n and for any integers a, b, “Euclidean algorithm” can calculate the non negative integer gcd(a, b). We have not formalized computational complexity of algorithms yet, though we had already formalize the “Euclidean algorithm” in [7]. For “right-to-left binary algorithm”, we formalize the theorem, which says that the required number of the modular squares and modular products in this algorithms are ⌊1+log2 n⌋ and for “Euclidean algorithm”, we formalize the Lamé’s theorem [6], which says the required number of the divisions in this algorithm is at most 5 log10 min(|a|, |b|). Our aim is to support the implementation of number theoretic tools and evaluating computational complexities of algorithms to prove the security of cryptographic systems.

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.

Turing reducibility and Turing degrees

2018 ◽  
Author(s):  
Harold Hodes
Keyword(s):  
Computational Complexity ◽  
Recursion Theory ◽  
Turing Degrees ◽  
Natural Numbers ◽  
Mathematical Objects ◽  
Classical Setting ◽  
Turing Reducibility

A reducibility is a relation of comparative computational complexity (which can be made precise in various non-equivalent ways) between mathematical objects of appropriate sorts. Much of recursion theory concerns such relations, initially between sets of natural numbers (in so-called classical recursion theory), but later between sets of other sorts (in so-called generalized recursion theory). This article considers only the classical setting. Also Turing first defined such a relation, now called Turing- (or just T-) reducibility; probably most logicians regard it as the most important such relation. Turing- (or T-) degrees are the units of computational complexity when comparative complexity is taken to be T-reducibility.

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.

The Addition of Primes and Power

1996 ◽  
Vol 48 (3) ◽  
pp. 512-526 ◽  
Author(s):  
Jörg Brüdern ◽  
Alberto Perelli
Keyword(s):  
Natural Number ◽  
Riemann Hypothesis ◽  
Natural Numbers

AbstractLet k ≥ 2 be an integer. Let Ek(N) be the number of natural numbers not exceeding N which are not the sum of a prime and a k-th power of a natural number. Assuming the Riemann Hypothesis for all Dirichlet L-functions it is shown that Ek(N) ≪ N1-1/25k.

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