The aim of this diploma work is to analyse problems solving of the second round of the 55th year of the Mathematical Olympiad in South Bohemian region and to serve as a study material for further Mathematical Olympiads or as a collection of problems for talented students.

Estudamos a criptografia RSA como uma importante aplicação dos números primos e da aritmética modular. Apresentamos algumas sugestões de atividades relacionadas ao tema a serem desenvolvidas em sala de aula nas séries finais do ensino fundamental / We studied RSA cryptography as an important application to prime numbers and modular arithmetic. We present some suggestions of activities related to the subject to be developed in classrooms of the final years of elementary school vii

Orientador: José Plínio de Oliveira Santos / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-27T11:40:03Z (GMT). No. of bitstreams: 1 Nascimento_MarcusViniciusSilva_M.pdf: 918557 bytes, checksum: de0f1627892732c764e7f5046966336f (MD5) Previous issue date: 2015 / Resumo: Nosso objetivo, nesse trabalho, é o de fazer um estudo sobre o método do crivo. A motivação reside no desejo de aplicar essas ideias a uma situação particular. Dividimos nosso trabalho em três partes. Na primeira fornecemos apenas as definições e con- ceitos básicos. Na segunda apresentamos o principio da inclusão-exclusão que embora sendo algo bastante conhecido merece destaque especial dada a sua importância como ferramenta no nosso trabalho. Na terceira e última parte, fazemos uma contextualização histórica e uma descrição da evolução das ideias do crivo de Eratóstenes-Legendre. A escolha desse crivo, dentre tantos outros, foi feita tendo em vista dois pontos. O primeiro é que o crivo de Eratóstenes-Legendre é o mais simples dentre os crivos estudados na teoria dos crivos. O segundo ponto está relacionado com o fato deste crivo fornecer a ideia geral dos crivos combinatoriais, uma vez que os crivos mais sofisticados são extensões de suas ideias básicas / Abstract: Our aim in this work is to make a study about the sieve method. The motivation lies in the intent of applying this idea in a particular situation. We splitted the study into three parts. The first part deals with definitions and basic concepts. In the second we present the principle of inclusion-exclusion while being something well known deserves special mention given its importance as a tool in our work. In the third and final part, we make a historical contextualization and a description of the evolution of the sieve Eratosthenes- Legendre ideas. The choice of sieve, among many others, has been made taking into account two points. The first is that the Eratosthenes-Legendre sieve is the simplest among the sieves studied the theory of sieves. The second point is related to the fact that this sieve provide the general idea of combinatorial sieve, since the more sophisticated sieves are extensions of its basic idea / Mestrado / Matematica Aplicada / Mestre em Matemática Aplicada

The Goldbach conjecture states that every even number greater than 2 can be written as a sumof two prime numbers. This thesis will go through the necessary theory and the backgroundto the problem at hand. Some numerical results connected to the Goldbach conjecture suchas displaying Goldbach partitions will be presented visually and interpretations of what theseresults yield will be made. How the Goldbach partitions behave for large even numbers will bestudied as well as patterns within these results. The tendencies in the graphical results supportthat the Goldbach conjecture could be true.

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The author has found, during her experience teaching students on the fourth grade level, that some concepts of number theory haven't even been introduced to the students. Some of these concepts include prime and composite numbers and their applications. Through personal research, the author has found that prime numbers are vital to the understanding of the grade level curriculum. Prime numbers are used to aide in determining divisibility, finding greatest common factors, least common multiples, and common denominators. Through experimentation, classroom examples, and homework, the author has introduced students to prime numbers and their applications.

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This paper is a bachelor thesis in Applied Mathematics at Linnaeus University. The goal of this thesis is to find a structure in sets of discrete quadratic dynamical systems modulo a number, with a main focus on 1-, and 2-periodic points in sets of systems modulo an odd prime. The amount of 1-, and 2-periodic points in such sets is numerically investigated and is proven directly to be p, respectively p-1. Furthermore, the sets are visualized using a diagram, where some apparent structures are noticed, and later explained. Finally, through numerical investigations, an expression for the amount of 2-periodic points in a system modulo a composite number is also perceived, and expressed as a conjecture. Proving the conjecture should be done in a future project.

Cette thèse concerne quelques problèmes liés à la répartition des entiers sans facteur carré dansles progressions arithmétiques. Ces problèmes s’expriment en termes de majorations du terme d’erreurassocié à cette répartition.Les premier, deuxième et quatrième chapitres sont concentrés sur l’étude statistique des termesd’erreur quand on fait varier la progression arithmétique modulo q. En particulier on obtient une formuleasymptotique pour la variance et des majorations non triviales pour les moments d’ordre supérieur. Onfait appel à plusieurs techniques de théorie analytique des nombres comme les méthodes de crible et lessommes d’exponentielles, notamment une majoration récente pour les sommes d’exponentielles courtesdue à Bourgain dans le deuxième chapitre.Dans le troisième chapitre on s’intéresse à estimer le terme d’erreur pour une progression fixée. Onaméliore un résultat de Hooley de 1975 dans deux directions différentes. On utilise ici des majorationsrécentes de sommes d’exponentielles courtes de Bourgain-Garaev et de sommes d’exponentielles torduespar la fonction de Möbius dues à Bourgain et Fouvry-Kowalski-Michel. / This thesis concerns a few problems linked with the distribution of squarefree integers in arithmeticprogressions. Such problems are usually phrased in terms of upper bounds for the error term relatedto this distribution.The first, second and fourth chapter focus on the satistical study of the error terms as the progres-sions varies modulo q. In particular we obtain an asymptotic formula for the variance and non-trivialupper bounds for the higher moments. We make use of many technics from analytic number theorysuch as sieve methods and exponential sums. In particular, in the second chapter we make use of arecent upper bound for short exponential sums by Bourgain.In the third chapter we give estimates for the error term for a fixed arithmetic progression. Weimprove on a result of Hooley from 1975 in two different directions. Here we use recent upper boundsfor short exponential sums by Bourgain-Garaev and exponential sums twisted by the Möbius functionby Bourgain et Fouvry-Kowalski-Michel.

Cryptography fascinates people of all generations and is increasingly presented as an example for the relevance and application of the mathematical sciences. Indeed, many principles of modern cryptography can be described at a secondary school level. In this context, the mathematical background is often only sparingly shown. In the worst case, giving mathematics this character of a tool reduces the application of mathematical insights to the message ”cryptography contains math”. This paper examines the question as to what else cryptography can offer to mathematics education. Using the RSA cryptosystem and related content, specific mathematical competencies are highlighted that complement standard teaching, can be taught with cryptography as an example, and extend and deepen key mathematical concepts.