scholarly journals OS USOS FLEXÍVEIS DO CONCEITO DE VARIÁVEL NA EDUCAÇÃO BÁSICA: UM ESTUDO COM AS EQUAÇÕES DIOFANTINAS LINEARES

2015 ◽  
Vol 37 ◽  
pp. 356
Author(s):  
Wagner Marcelo Pommer
Keyword(s):  
Content Analysis ◽  
Diophantine Equation ◽  
Functional Relationship ◽  
Diophantine Equations ◽  
Basic Education ◽  
Linear Equations ◽  
Linear Diophantine Equation ◽  
Linear Diophantine Equations ◽  
Theoretical Considerations ◽  
Literal Symbols

http://dx.doi.org/10.5902/2179460X14370In many curricular documents for Basic Education the variable concept is briefly quoted, seen as a paramathematics notion, intrinsic to the development of functions, without further references. This article aims to present and discuss the context of Linear Diophantine Equation as a possible theme to explore the flexible uses of the variable concept in algebraic education. The theoretical considerations is based upon Küchemann (1981), Usiskin (1995) and Ursini;Trigueros (2001), researchers who believe that variables can assume different roles: unknown, generalized number and in a functional relationship. In that scenary, the understanding of the variable concept pervades some potential ways. We were inspired at Content Analysis, described in Bardin (2004), as metodological referencial to search contexts that favour such framework. In preanalisys realized we consider the epistemology present on Diophantine Linear Equations theme as a possible way to explore the flexible uses of the variable concept. The analysis revealed that the Linear Diophantine Equations allow the acquisition of the following potentialities, expressed in Ursini and Trigueiros (2001): executing calculations and simple operations with the literal symbols; create a context of integration to the variables flexible uses; allow situating some advantages on the different uses of the variable concept.

AN APPLICATION OF LINEAR DIOPHANTINE EQUATIONS TO CRYPTOGRAPHY

2021 ◽  
Vol 10 (6) ◽  
pp. 2799-2806
Author(s):  
P. Anuradha Kameswari ◽  
S.S. Sriniasarao ◽  
A. Belay
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Key Exchange ◽  
Infinitely Many Solutions ◽  
Key Exchange Protocol ◽  
Random Solution ◽  
Linear Diophantine Equation ◽  
Linear Diophantine Equations

In this chapter we propose a Key exchange protocol based on a random solution of linear Diophantine equation in n variables, where the considered linear Diophantine equation satisfies the condition for existence of infinitely many solutions. Also the crypt analysis of the protocol is analysed.

scholarly journals On Infinite Number of Solutions for one type of Non-Linear Diophantine Equations

2019 ◽  
Vol 9 (1) ◽  
pp. 1665-1669
Keyword(s):  
Infinite Number ◽  
Diophantine Equation ◽  
Diophantine Equations ◽  
Prime Numbers ◽  
Periodic Sequence ◽  
Number Of Solutions ◽  
Linear Diophantine Equation ◽  
Linear Diophantine Equations ◽  
Non Linear

In this article, we prove that the non-linear Diophantine equation 𝑦 = 2𝑥1𝑥2 …𝑥𝑘 + 1; 𝑘 ≥ 2, 𝑥𝑖 ∈ 𝑃 − {2}, 𝑥𝑖′𝑠 are distinct and P is the set of all prime numbers has an infinite number of solutions using the notion of a periodic sequence. Then we also obtained certain results concerning Euler Mullin sequence.

scholarly journals FRAÇÕES CONTÍNUAS, DETERMINANTES E EQUAÇÕES DIOFANTINAS LINEARES

2015 ◽  
Vol 37 ◽  
pp. 95
Author(s):  
Delfim Dias Bonfim ◽  
Gilmar Pires Novaes
Keyword(s):  
Diophantine Equation ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Diophantine Equations ◽  
Geometric Interpretation ◽  
Linear Diophantine Equation ◽  
Linear Diophantine Equations ◽  
Definition Of ◽  
Fundamental Theorems

http://dx.doi.org/10.5902/2179460X14468This article is intended to present a method for solving linear diophantine equations, using for this purpose, the concepts of continuous fractions and determinants. Initially we present the definition of simple continued fraction, geometric interpretation and some fundamental theorems related to this concept. Subsequently we relate the finite simple continued fractions with determinants. Finally we present the definition of linear Diophantine equation and we demonstrate the method to solve it using the concepts mentioned above.

scholarly journals Matrix Diophantine equations over quadratic rings and their solutions

2020 ◽  
Vol 12 (2) ◽  
pp. 368-375
Author(s):  
N.B. Ladzoryshyn ◽  
V.M. Petrychkovych ◽  
H.V. Zelisko
Keyword(s):  
Matrix Equation ◽  
Diophantine Equations ◽  
Linear Equations ◽  
Standard Form ◽  
Matrix Equations ◽  
System Of Linear Equations ◽  
Structure Of Solutions ◽  
Linear Diophantine Equations ◽  
The Matrix ◽  
Solution Methods

The method for solving the matrix Diophantine equations over quadratic rings is developed. On the basic of the standard form of matrices over quadratic rings with respect to $(z,k)$-equivalence previously established by the authors, the matrix Diophantine equation is reduced to equivalent matrix equation of same type with triangle coefficients. Solving this matrix equation is reduced to solving a system of linear equations that contains linear Diophantine equations with two variables, their solution methods are well-known. The structure of solutions of matrix equations is also investigated. In particular, solutions with bounded Euclidean norms are established. It is shown that there exists a finite number of such solutions of matrix equations over Euclidean imaginary quadratic rings. An effective method of constructing of such solutions is suggested.

scholarly journals Linear diophantine equations with cyclic coefficient matrices and its applications to Riemann surfaces

1985 ◽  
Vol 28 (3) ◽  
pp. 369-380
Author(s):  
Nobumasa Takigawa
Keyword(s):  
Riemann Surfaces ◽  
Diophantine Equations ◽  
Linear Equations ◽  
Integral Solution ◽  
Complex Numbers ◽  
Necessary And Sufficient Condition ◽  
Linear Diophantine Equations ◽  
Fixed Complex ◽  
Image Position ◽  
Necessary And Sufficient

Let co, c1, …, cn-1 be the nonzero complex numbers and let C = (cu+1,v+1) = (cn+u-v), O≦u,v≦n — 1, be a cyclic matrix, where n + u — v is taken modulo n. In this paper we shall give the solution of the linear equationswhere Lu (0≦u≦n —1) is a fixed complex number. In Theorem 1 weshall give a necessary and sufficient condition for (1) to have an integral solution.

scholarly journals MÉTODO PARA RESOLVER EQUAÇÕES DIOFANTINAS COM COEFICIENTES NO CONJUNTO DOS NÚMEROS RACIONAIS

2015 ◽  
Vol 37 ◽  
pp. 47
Author(s):  
Josias Neubert Savóis ◽  
Daiane Freitas
Keyword(s):  
Diophantine Equations ◽  
Basic Education ◽  
Daily Life ◽  
Rational Numbers ◽  
Greatest Common Divisor ◽  
Practical Application ◽  
Mathematical Concepts ◽  
The Real ◽  
Linear Diophantine Equations ◽  
The Creation

http://dx.doi.org/10.5902/2179460X14529Develop a solid understanding of linear Diophantine equations in two variables facilitates the resolution of many problems of daily life and also the real understanding of some mathematical concepts taught in school but they seem useless and without practical application. Moreover, the relationships that these equations are established with other content that are already inserted into the basic education justify their education and become an important tool for contextualizing and interdisciplinarity. This work also aims to show the importance of teaching of rational numbers and in this context do analysis on a new concept of greatest common divisor,called generalized maximum common divisor, and thus we can use the numberpad as rational coefficients of Diophantine equations, expanding the breadth of problems solved by these equations. The creation of various practical problems of implementation of the theory studied serves to convince us of the importance of this work and to encourage their implementation and creating new problems taking into account the reality of each school and its students.

scholarly journals The Mathematical Safe Problem and Its Solution (Part 2)

2021 ◽  
pp. 16-28
Author(s):  
Sergii Kryvyi ◽  
Hryhorii Hoherchak
Keyword(s):  
Commutative Ring ◽  
Finite Fields ◽  
Computer Games ◽  
Diophantine Equations ◽  
Linear Equations ◽  
Finite Rings ◽  
Prime Field ◽  
Linear Diophantine Equations ◽  
Systems Of Linear Equations ◽  
Mathematical Safe

Introduction. The problem of mathematical safe arises in the theory of computer games and cryptographic applications. The article considers numerous variations of the mathematical safe problem and examples of its solution using systems of linear Diophantine equations in finite rings and fields. The purpose of the article. To present methods for solving the problem of a mathematical safe for its various variations, which are related both to the domain over which the problem is considered and to the structure of systems of linear equations over these domains. To consider the problem of a mathematical safe (in matrix and graph forms) in different variations over different finite domains and to demonstrate the work of methods for solving this problem and their efficiency (systems over finite simple fields, finite fields, ghost rings and finite associative-commutative rings). Results. Examples of solving the problem of a mathematical safe, the conditions for the existence of solutions in different areas, over which this problem is considered. The choice of the appropriate area over which the problem of the mathematical safe is considered, and the appropriate algorithm for solving it depends on the number of positions of the latches of the safe. All these algorithms are accompanied by estimates of their time complexity, which were considered in the first part of this paper. Conclusions. The considered methods and algorithms for solving linear equations and systems of linear equations in finite rings and fields allow to solve the problem of a mathematical safe in a large number of variations of its formulation (over finite prime field, finite field, primary associative-commutative ring and finite associative-commutative ring with unit). Keywords: mathematical safe, finite rings, finite fields, method, algorithm.

Secure Watermarking using Diophantine Equations for Authentication and Recovery

2015 ◽  
Vol 3 (2) ◽  
Author(s):  
Jayashree Nair ◽  
T. Padma
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Jpeg Compression ◽  
Authentication Scheme ◽  
Original Image ◽  
Invariant Features ◽  
Jpeg2000 Compression ◽  
Frequency Properties ◽  
Visual Authentication

This paper describes an authentication scheme that uses Diophantine equations based generation of the secret locations to embed the authentication and recovery watermark in the DWT sub-bands. The security lies in the difficulty of finding a solution to the Diophantine equation. The scheme uses the content invariant features of the image as a self-authenticating watermark and a quantized down sampled approximation of the original image as a recovery watermark for visual authentication, both embedded securely using secret locations generated from solution of the Diophantine equations formed from the PQ sequences. The scheme is mildly robust to Jpeg compression and highly robust to Jpeg2000 compression. The scheme also ensures highly imperceptible watermarked images as the spatio –frequency properties of DWT are utilized to embed the dual watermarks.

Sign in / Sign up

Export Citation Format

Share Document