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# An Introductory Course in Elementary Number Theory

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... There are also many concepts the continent needs to retrieve it, such as the Chinese remainder theorem and the multiplication inverse [9]. For multivariate functions, the concept is related to its derivation. ...
... then is said relatively prime [8]. 2579 The function f is said multiplicative if f(xy)=f(x)f(y) and gcd(x,y)=1 [9]. ...
... Multiplicative function phi is said Euler's phi if gcd(x,y)=1 then phi(xy)=phi(x)phi(y) [9]. ...
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Many researchers have studied the development and updating of RSA method and methods with the key, where these methods depend on the powers and confidentiality of the keys. In our research, we worked on these keys, which are numbers and for confidentiality we treated them as polynomial where the recipient of the cipher worked to solve them and return them as numbers. These polynomials were built with specific theories based on partial differential equations where they gave positive results achieved through examples.
... There are also many concepts the continent needs to retrieve it, such as the Chinese remainder theorem and the multiplication inverse [9]. For multivariate functions, the concept is related to its derivation. ...
... then is said relatively prime [8]. 2579 The function f is said multiplicative if f(xy)=f(x)f(y) and gcd(x,y)=1 [9]. ...
... Multiplicative function phi is said Euler's phi if gcd(x,y)=1 then phi(xy)=phi(x)phi(y) [9]. ...
Article
Full-text available
Many researchers have studied the development and updating of RSA method and methods with the key, where these methods depend on the powers and confidentiality of the keys. In our research, we worked on these keys, which are numbers and for confidentiality we treated them as polynomial where the recipient of the cipher worked to solve them and return them as numbers. These polynomials were built with specific theories based on partial differential equations where they gave positive results achieved through examples.
... Also, from the above example it is obtained that if (n = p) is prime, then ϕ (p) = p − 1 and in general, ϕ p k = p k − p k−1 for any positive integer k. For any positive integer n we have that ϕ (n) = n 1 − 1 [1,4,6,8,9]. ...
... A function f is said to be multiplicative if for all positive integers m, n such that m, n are relatively prime, then f (mn) = f (m) f (n). Both the Euler ϕ− function and the Mobius function are multiplicative [8]. ...
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Let G = (V, E) be a simple connected undirected graph. In this paper, we define generalized Euler's Φ-Function of a graph which is the summation of the Euler's ϕ−function of the degree of the vertices of a graph and it is denoted by Φ (G). It is determined the general form of Euler's Φ-function of some standard graphs. Finally, some important results and properties are studied.
... Also, from the above example it is obtained that if (n = p) is prime, then ϕ (p) = p − 1 and in general, ϕ p k = p k − p k−1 for any positive integer k. For any positive integer n we have that ϕ (n) = n 1 − 1 [1,4,6,8,9]. ...
... A function f is said to be multiplicative if for all positive integers m, n such that m, n are relatively prime, then f (mn) = f (m) f (n). Both the Euler ϕ− function and the Mobius function are multiplicative [8]. ...
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... From the definition of complete residue systems, [Raj13] definition 13, V forms a complete residue system modulo b. As a, b are coprime, from [Raj13] Theorem 24, it follows that W forms a complete residue system modulo b. ...
... From the definition of complete residue systems, [Raj13] definition 13, V forms a complete residue system modulo b. As a, b are coprime, from [Raj13] Theorem 24, it follows that W forms a complete residue system modulo b. ...
Preprint
This paper presents a closed form polynomial expression for the binary cyclotomic polynomial. We contrast this against expressions for binary cyclotomic polynomials in (Lam and Leung 1996) and (Lenstra 1979).
... Used since ancient times in arithmetic, numbers are part and parcel of pure mathematics and fundamental to the study of integers. Numbers theory, sometimes also called the "queen of mathematics," is thus the branch of mathematics that encompasses the properties and relationships of numbers, especially positive integers (Katz, 2014;Raji, 2013). ...
Chapter
Encryption is the process of converting confidential private data into unreadable form and securing information in the file from unauthorized access using various encryption algorithms. We live in the information age where the exchange of private information has become the integral part of our day-to-day activities. Billions of e-mails and business data are sent throughout the world through internet daily. The success of the information age is to keep private secure data from unauthorized access and key to access the private and secure data for authorized users. Encryption in this information age plays a vital role in the protecting the confidential data from unauthorized access. In the last few decades, the computer network has created a revolution in the use of information. Authorized users access their data or send their private data from anywhere in the world; hence, it has become very important to secure the private data not only where it is stored, but also to maintain high level of confidentiality while transmission of this private data from one machine to another.
... Prinsip pertama induksi matematika adalah if a set of positive integers has the property that, if it contains the integer k, then it also contains k + 1, and if this set contains 1 then it must be the set of all positive integers (Raji, 2013). ...
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Students' ability in solving argumentation in mathematical induction and binomial theory is still lacking based on the results of the exam, so that evidence-based teaching materials are prepared. This study uses design research methods with the subject from 2nd semester students of the Mathematics Education Study Program. This research is a development research consisting of 6 stages: Identification of needs, Planning, Initial product development, Initial product testing, Product revision, Field trials so as to produce teaching materials. The results of the lecturer and student validation show that the teaching materials are valid and feasible to use with an average score of 45 and 45,9333. While the test of the effectiveness of teaching materials used using the t-test obtained results that ttest> ttable, 5.83 > 2.11 which means there is an increase in the average value of students before and after using teaching materials. With the increase in value, it indicates that the instructional materials prepared are effectively used.
... No que concerne ao primeiro item, registra-se a predominância dos livros de HM consultados atribuírem a autoria da formulação (**), descoberta pelo matemático francês, em 1843, Jacques-Phillipe-Marie Binet (1786-1856). Entretanto, Koshy (2011) Para sua validação, Brousseau (1965), Huntley (1970) e Raji (2015) empregam o modelo de Indução Matemática. Possivelmente, o mesmo recebe a preferência da maior parte dos autores consultados aqui. ...
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Este trabalho apresenta uma proposta de Engenharia Didática - ED que descreve, apenas, as etapas de análises preliminares, análise a priori (e a concepção de duas situações didáticas), que correspondem aos dois primeiros momentos previstos, de modo sistemático, pela ED no campo da Didática da Matemática. O tema envolve uma das formas ou possibilidades de generalização do modelo secular, que prevê a reprodução de casais de coelhos, nomeado por Sequência de Fibonacci – SF. Diante do rico e variado percurso de evolução e sistematização do referido modelo, opta pela generalização de uma fórmula, cuja autoria, envolve informações conflitantes, embora seja mais conhecida, de acordo com os autores de livros de História da Matemática, como fórmula de Binnet ou Teorema de Binnet. A descrição de duas etapas iniciais de uma ED, envolvem elementos que detêm a possibilidade de explorar novas concepções dos estudantes acerca da SF, inclusive, tendo em vista o uso do CAS Maple, num contexto de investigação histórica, assumindo uma perspectiva de ensino afetada pela Teoria das Situações Didáticas – TSD.AbstractThis work presents a proposal of Didactical Engineering – DE, describing just the preliminarly analyzes and a priori analysis (and the didactical conception´s situations) which correspond to the first two levels in accordance to ED, in the Didactics of Mathematics. The theme involves one way of possibilities to obtain a kind of generalization relative to a secular model, which provides the couples of rabbits reproduction, namely by Fibonacci Sequence – FS. Before the rich and varied path of evolution and its systematization of this model, it is opted for the formula generalization, whose authorship involves a conflicting information, however, it is known, according to the authors of Mathematical History, as a Binnet´s formula or the Binnet´s theorem. The description of the two initial research steps involves some elements that have the potencial to explore new conceptions of the students about the FS, even considering the use of CAS Maple, in a historical research context, supported by the Theory of Didactical Situations – TDS.Keywords: Fibonacci Sequence. Model Generalization. Didactical Engineering. Teaching and Technology.
Article
There have been several fascinating applications of Number Theory in key cryptography. Key cryptography enables many technologies we take for granted, such as the ability to make secure online transactions. The purpose of this survey paper is to highlight certain important such applications. Prime numbers constitute an interesting and challenging area of research in number theory. Diophantine equations form the central part of number theory. An equation requiring integral solutions is called a Diophantine equation. In the first part of this paper, some major contribution in number theory using prime number theorem is discussed and some of the problems which still remains unsolved are covered. In the second part some of the theorems and functions are also discussed such as Diophantine Equation, Goldbach conjecture, Fermats Theorem, Riemann zeta function and his hypothesis that still remain unproved to this day . The Chinese hypothesis is a special case of Fermat’s little theorem. As proved later in the west, the Chinese hypothesis is only half right . From the data of this study we conclude that number theory is used in computer network and applications in cryptography. We came to know about the purpose of Diophantine equation , Square-free natural number, Zeta function, Fermat’s theorem and Chinese hypothesis which is a special case for Fermat’s theorem.
Article
In this paper, we deal with specific cases in which we can analyze the code of Shimada based on the status of sending the same plaintext from more than one user with different public keys.Wewere able to analyze this codewith the help of the Chinese remainder theorem.
Chapter
For some purposes one requires to know not merely that a lattice Λ has a point in a set L, but that it has a number of linearly independent points in L.
Book
From the contents: Primes in Arithmetic Progression.- Gauss' Sum.- Cyclotomy.- Primes in Arithmetic Progression: The General Modulus.- Primitive Characters.- Dirichlet's Class Number Formula.- The Distribution of the Primes.- Riemann's Memoir.- The Functional Equation of the L Function.- Properties of the Gamma Function.- Integral Functions of Order 1.- The Infinite Products for xi(s) and xi(s,Zero-Free Region for zeta(s).- Zero-Free Regions for L(s, chi).- The Number N(T).- The Number N(T, chi).- The explicit Formula for psi(x).- The Prime Number Theorem.- The Explicit Formula for psi(x,chi).- The Prime Number Theorem for Arithmetic Progressions (I).- Siegel's Theorem.- The Prime Number Theorem for Arithmetic Progressions (II).- The Polya-Vinogradov Inequality.- Further Prime Number Sums.
Book
We owe to MINKOWSKI the fertile observation that certain results which can be made almost intuitive by the consideration of figures in n-dimensional euclidean space have far-reaching consequences in diverse branches of number theory. For example, he simplified the theory of units in algebraic number fields and both simplified and extended the theory of the approximation of irrational numbers by rational ones (Diophantine Approximation). This new branch of number theory, which MINKOWSKI christened “The Geometry of Numbers”, has developed into an independent branch of number-theory which, indeed, has many applications elsewhere but which is well worth studying for its own sake.
Article
Now into its Eighth edition, The Higher Arithmetic introduces the classic concepts and theorems of number theory in a way that does not require the reader to have an in-depth knowledge of the theory of numbers The theory of numbers is considered to be the purest branch of pure mathematics and is also one of the most highly active and engaging areas of mathematics today. Since earlier editions, additional material written by J. H. Davenport has been added, on topics such as Wiles' proof of Fermat's Last Theorem, computers & number theory, and primality testing. Written to be accessible to the general reader, this classic book is also ideal for undergraduate courses on number theory, and covers all the necessary material clearly and succinctly.
Article
It is well known that one can obtain explicit continued fraction expansions of ez for various interesting values of z ; but the details of appropriate constructions are not widely known. We provide a reminder of those methods and do that in a way that allows us to mention a number of techniques generally useful in dealing with continued fractions. Moreover, we choose to consider some expansions in Gaussian integers, allowing us to display some new results and to indicate some generalisations of classical results.