Esta dissertação tem por objetivo apresentar a Criptografia RSA, que é o método de criptografia mais utilizado no mundo atualmente. Iniciamos a dissertação com um breve histórico sobre a criptografia e em seguida introduzimos a teoria matemática empregada no método pertencente a teoria dos números. Finalizamos a dissertação com a descrição de uma aplicação simples do método levado para uma sala de aula do ensino médio. Este texto pretende introduzir o tema de maneira simples e por esta razão, fazemos uso de muitos exemplos. Esperamos ainda que o leitor compreenda o que torna este método eficiente e seguro. / The main goal of this work is to introduce the RSA Criptography that is the most used method in Criptography nowadays. We begin the dissertation with a brief introduction about criptography and then we discuss concepts from number theory used in the method. Finally we present a description of a simple application of Criptography made in a High school classroom. This text intend to introduce the subject in a simple way for this reason we present several examples. We hope that the reader have the comprehension of the methods and of its security.

Nesta dissertação são apresentadas algumas propriedades de números reais. Descrevemos de maneira breve os conjuntos numéricos N, Z, Q e R e apresentamos demonstrações detalhadas da irracionalidade dos números \'pi\' e e. Também, apresentamos um texto sobre o número e, menos técnico e mais intuitivo, na tentativa de auxiliar o professor no preparo de aulas sobre o número e para alunos do Ensino Médio, bem como, alunos de cursos de Licenciatura em Matemática / In this thesis we present some properties of real numbers. We describe briefly the numerical sets N, Z, Q and R, and we present detailed proofs of irrationality of numbers \'pi\' and e. We also present a text about the number e less technical and more intuitive in an attempt to assist the teacher in preparing lessons about number e for High School students as well as for Teaching degree in Mathematics students

Neste trabalho, exploramos os números de Fibonacci e de Lucas. A maioria dos resultados históricos sobre esses números são apresentados e provados. Ao longo do texto, um grande número de identidades a respeito dos números de Fibonacci e de Lucas são mostradas válidas para todos os inteiros. Sequências generalizadas de Fibonacci, a relação entre os números de Fibonacci e de Lucas com as raízes da equação x2 -x -1 = 0 e a conexão entre os números de Fibonacci e de Lucas com uma classe de matrizes em M2(R) são também exploradas. / In this work we explore the Fibonacci and Lucas numbers. The majority of the historical results are stated and proved. Along the text several identities concerning Fibonacci and Lucas numbers are shown valid for all integers. Generalized Fibonacci sequences, the relation between Fibonacci and Lucas numbers with the roots of the equation x2 -x -1 = 0 and the connection between Fibonacci and Lucas numbers with a class of matrices in M2(R) are also explored.

This thesis is primarily concerned in studying the relationship between different exponents of Diophantine approximation, which are quantities arising naturally in the study of rational approximation to a fixed n-tuple of real irrational numbers. As Khinchin observed, these exponents are not independent of each other, spurring interest in the study of the spectrum of a given family of exponents, which is the set of all possible values that can be taken by said family of exponents. Introduced in 2009-2013 by Schmidt and Summerer and completed by Roy in 2015, the parametric geometry of numbers provides strong tools with regards to the study of exponents of Diophantine approximation and their associated spectra by the introduction of combinatorial objects called n-systems. Roy proved the very surprising result that the study of spectra of exponents is equivalent to the study of certain quantities attached to n-systems. Thus, the study of rational approximation can be replaced by the study of n-systems when attempting to determine such spectra. Recently, Roy proved two new results for the case n=3, the first being that spectra are semi-algebraic sets, and the second being that spectra are stable under the minimum with respect to the product ordering. In this thesis, it is shown that both of these results do not hold in general for n>3, and examples are given. This thesis also provides non-trivial examples for n=4 where the spectra is stable under the minimum. An alternate and much simpler proof of a recent result of Marnat-Moshchevitin proving an important conjecture of Schmidt-Summerer is also given, relying only on the parametric geometry of numbers instead. Further, a conjecture which generalizes this result is also established, and some partial results are given towards its validity. Among these results, the simplest, but non-trivial, new case is also proven to be true. In a different vein, this thesis considers certain generalizations theta(q) of the classical theta q-series. We show under conditions on the coefficients of the series that theta(q) is neither rational nor quadratic irrational for each integer q>1.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 1981. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND HUMANITIES. / Bibliography: leaves 205-206. / by Fredric M. Katz. / Ph.D.

Ramsey Theorem, in the most simple form, states that if we are given a positive integer l, there exists a minimal integer r(l), called the Ramsey number, such any partition of the edges of K_r(l) into two sets, i.e. a 2-coloring, yields a copy of K_l contained entirely in one of the partitioned sets, i.e. a monochromatic copy of Kl. We prove an extension of Ramsey's Theorem, in the more general form, by replacing complete graphs by multipartite graphs in both senses, as the partitioned set and as the desired monochromatic graph. More formally, given integers l and k, there exists an integer p(m) such that any 2-coloring of the edges of the complete multipartite graph K_p(m);r(k) yields a monochromatic copy of K_m;k . The tools that are used to prove this result are the Szemeredi Regularity Lemma and the Blow Up Lemma. A full proof of the Regularity Lemma is given. The Blow-Up Lemma is merely stated, but other graph embedding results are given. It is also shown that certain embedding conditions on classes of graphs, namely (f , ?) -embeddability, provides a method to bound the order of the multipartite Ramsey numbers on the graphs. This provides a method to prove that a large class of graphs, including trees, graphs of bounded degree, and planar graphs, has a linear bound, in terms of the number of vertices, on the multipartite Ramsey number.

The notion of a normal number and the Normal Number Theorem date back over 100 years. Émile Borel first stated his Normal Number Theorem in 1909. Despite their seemingly basic nature, normal numbers are still engaging many mathematicians to this day. In this paper, we provide a reinterpretation of the concept of a normal number. This leads to a new proof of Borel's classic Normal Number Theorem, and also a construction of a set that contains all absolutely normal numbers. We are also able to use the reinterpretation to apply the same definition for a normal number to any point in a symbolic dynamical system. We then provide a proof that the Fibonacci system has all of its points being normal, with respect to our new definition. / Graduation date: 2012

This thesis reports on a longitudinal study of students’ understanding of decimal notation. Over 3000 students, from a volunteer sample of 12 schools in Victoria, Australia, completed nearly 10000 tests over a 4-year period. The number of tests completed by individual students varied from 1 to 7 and the average inter-test time was 8 months. The diagnostic test used in this study, (Decimal Comparison Test), was created by extending and refining tests in the literature to identify students with one of 12 misconceptions about decimal notation. (For complete abstract open document)

Thesis (M.A.)--University of North Carolina Wilmington, 2008. / Title from PDF title page (viewed May 26, 2009) Includes bibliographical references (p. 64-72)

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-02-11T11:30:44Z No. of bitstreams: 1 jorgemagestedacruzjunior.pdf: 2261526 bytes, checksum: 2e39ac93f53b7c28ef8a81bcdcd222af (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-02-26T11:58:03Z (GMT) No. of bitstreams: 1 jorgemagestedacruzjunior.pdf: 2261526 bytes, checksum: 2e39ac93f53b7c28ef8a81bcdcd222af (MD5) / Made available in DSpace on 2016-02-26T11:58:03Z (GMT). No. of bitstreams: 1 jorgemagestedacruzjunior.pdf: 2261526 bytes, checksum: 2e39ac93f53b7c28ef8a81bcdcd222af (MD5) Previous issue date: 2014-04-22 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O presente trabalho tem por objetivo descrever e conceituar a importância dos números áureos. Sua aplicabilidade acompanha o ser humano e frequentemente são vivenciados em situações cotidianas. Durante a elaboração deste estudo procurou-se demonstrar as diferentes aparições do número áureo, nas mais diversas áreas em que vivemos, seja na natureza, nos animais, na arquitetura e até mesmo no corpo humano. A pesquisa foi realizada através de consultas em livros escritos por autores renomados e em artigos publicados em bases de dados confiáveis. Esta pesquisa visa ampliar o conhecimento e apresentar aos alunos uma maneira diferente de ver e entender a matemática e sua aplicabilidade e influência no dia-a-dia. / The present work aims to describe and conceptualize the importance of golden numbers. Its applicability with humans and often are experienced in everyday situations. During the preparation of this study sought to demonstrate the different appearances of the Golden number, in the most diverse areas in which we live, whether in nature, animals and even in the human body. The survey was conducted through consultations in books written by renowned authors and in articles published in reliable databases. This research aims to expand the knowledge and present to students a different way to see and understand the mathematics and its applicability and influence in everyday life.