Este trabalho descreve as soluções de algumas equações diofantinas em duas e três variáveis. O objetivo é apresentar a análise de alguns casos simples e de outros mais difíceis relativos ao Último Teorema de Fermat. Primeiramente são apresentados os pré-requisitos necessários dentre os quais incluímos a noção de número primo, máximo divisor comum, congruência, o Algoritmo de Euclides e o Teorema Fundamental da Aritmética. Este material é desenvolvido primeiramente no anel dos inteiros racionais e posteriormente em duas extensões algébricas conhecidas como os inteiros de Gauss e de Eisenstein. A estrutura dos últimos é indispensável na resolução do primeiro caso não trivial do Último Teorema de Fermat, a saber, da equação diofantina x3 + y3 = z3. O último capítulo apresenta algumas aplicações de problemas diofantinos e do Algoritmo de Euclides que podem ser desenvolvidos em sala de aula com alunos do sexto e do oitavo ano. / This work describes the solutions to some diophantine equations in two and three variables. The objective is to present the analysis of some simple and other more difficult cases related to Fermats Last Theorem. First, we present the necessary prerequisites which include the notion of a prime number, the maximum common divisor, congruences, Euclids Algorithm and the Fundamental Theorem of Arithmetic. This material is first developed by using the rational integers and then presented for two algebraic extensions known as Gauss and Eisenstein integers. The structure of the latter is indispensable for the first non-trivial case of Fermats Last Theorem, namely, the diophantine equation x3 + y3 = z3. The last chapter presents some applications of simple diophantine equations and Euclids algorithm which can be developed in the classroom with sixth and eight grade students.

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.

Seja p um número primo e seja uma raiz p - ésima primitiva da unidade. Considere os seguintes elementos i := 1 + + 2 + ... + i-1 para todo 1 i k do anel Z[] onde k = (p-1)/2. Nesta tese nós descrevemos explicitamente um conjunto gerador para o grupo das unidades do anel de grupo integral ZC2p; representado por U(ZC2p); onde C2p representa o grupo cíclico de ordem 2p e p satisfaz as seguintes condições: S := { -1, , u2, ... uk } gera U(Z[]) e U(Zp) = ou U(Zp)2 = e -1 U(Zp); que são verificadas para p = 7; 11; 13; 19; 23; 29; 53; 59; 61 e 67. Com o intuito de estender tais ideias encontramos um conjunto gerador para U(Z(C2p x C2) e U(Z(C2p x C2 x C2) onde p satisfaz as mesmas condições anteriores acrescidas de uma nova hipótese. Finalmente com o auxílio dos resultados anteriores apresentamos um conjunto gerador das unidades centrais do anel de grupo Z(Cp x Q8); onde Q8 representa o grupo dos quatérnios, ou seja, Q8 := <a; b : a4 = 1; a2 = b2; b-1 a b = a-1 >. / Let p be an odd prime integer, be a pth primitive root of unity, Cn be the cyclic group of order n, and U(ZG) the units of the Integral Group Ring ZG: Consider ui := 1++2 +: : :+i1 for 2 i p + 1 2 : In our study we describe explicitly the generator set of U(ZC2p); where p is such that S := f1; ; u2; : : : ; up1 2 g generates U(Z[]) and U(Zp) is such that U(Zp) = 2 or U(Zp)2 = 2 and 1 =2 U(Zp)2; which occurs for p = 7; 11; 13; 19; 23; 29; 37; 53; 59; 61, and 67: For another values of p we don\'t know if such conditions hold. In addition, under suitable hypotheses, we extend these ideas and build a generator set of U(Z(C2p C2)) and U(Z(C2p C2 C2)): Besides that, using the previous results, we exhibit a generator set for the central units of the group ring Z(Cp Q8) where Q8 represents the quaternion group.

No description available.

Orientador: Andréia Cristina Ribeiro / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-17T16:44:00Z (GMT). No. of bitstreams: 1 Mucelin_Claudio_M.pdf: 744549 bytes, checksum: 062211ac0a3abf9bcf171fe9881dcafa (MD5) Previous issue date: 2011 / Resumo: Este trabalho apresenta alguns resultados sobre partições de números inteiros e a importância deles na história da Matemática e da Teoria dos Números. Encontrar demonstrações bijetivas em partições não é nada fácil. Mas, depois de encontradas, tornam-se uma maneira agradável e fácil de entender e provar algumas Identidades de Partições. Este trabalho pretende ser didático e de fácil entendimento para futuras pesquisas de estudantes que se interessem pelo assunto. Ele traz definições básicas e importantes sobre partições, os Gráficos de Ferrers, demonstrações de resultados interessantes como a Bijeção de Bressoud e o Teorema Pentagonal de Euler. Destaca também a importância das funções geradoras e alguns resultados devidos a Sylvester, Dyson, Fine, Schur e Rogers-Ramanujan / Abstract: This work presents some results about partitions of integers numbers and their importance in the history of Mathematics and in the Theory of the Numbers. To find bijective demonstrations in partitions it is not easy. But, after finding them, to understand and to prove some Identities of Partitions becomes agreeable and easy. This work intends to be didatic and of easy understanding for future researches made by students interested in this subject. It contains basic and important definitions about partitions, the Ferrers' Graphics, demonstrations of interesting results as the Bressond's Bijection and the Euler's Pentagonal Theorem. It also details the importance of the generating functions and some results due to Sylvester, Dyson, Fine, Schur and Rogers-Ramanujan / Mestrado / Teoria dos Numeros / Mestre em Matemática

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2015-10-22T13:38:57Z No. of bitstreams: 2 Dissertação - Lourival Carlos Cunha Junior - 2015.pdf: 8702813 bytes, checksum: 307cfb19927718cfa7a667f109f4768f (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-10-22T13:42:02Z (GMT) No. of bitstreams: 2 Dissertação - Lourival Carlos Cunha Junior - 2015.pdf: 8702813 bytes, checksum: 307cfb19927718cfa7a667f109f4768f (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-10-22T13:42:02Z (GMT). No. of bitstreams: 2 Dissertação - Lourival Carlos Cunha Junior - 2015.pdf: 8702813 bytes, checksum: 307cfb19927718cfa7a667f109f4768f (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-05-25 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This dissertation aims to analyze how the use of mathematical tricks, educational games and alternative algorithms for multiplication and Division of integers contribute to learning of multiplication and Division operations between positive integers. Research participants were students of a class of adult and youth education of the 2o segment of the Central Penitentiary Progression of Distrito Federal (CPP-DF). The analysis of the results was made from questionnaires answered by the students and observations by the teacher during class. The introduction of mathematical operations took place initially by mathematical tricks, then were presented alternative ways to perform them later manipulation of new algorithms, unraveling the tricks and finally organizing with mathematical games. The participants demonstrated commitment, dedication and interest in the tricks, games and alternative algorithms. The objective of this work was achieved. We realized that there was in fact an improvement in pupils ’ learning. / Esta dissertação tem como objetivo analisar como o uso de truques matemáticos, jogos pedagógicos e algoritmos alternativos para a multiplicação e divisão de números inteiros contribuem para o aprendizado das operações multiplicação e divisão entre números inteiros positivos. Os participantes da pesquisa foram alunos de uma turma de Educação de Jovens e Adultos do 2o Segmento do Centro de Progressão Penitenciária do Distrito Federal (CPP-DF). A análise dos resultados foi feita a partir de questionários respondidos pelos alunos e observações realizadas pelo professor durante as aulas. A introdução das operações matemáticas ocorreu inicialmente por truques matemáticos, em seguida foram apresentadas formas alternativas de realizá-las, posteriormente a manipulação dos novos algoritmos, desvendando os truques e finalmente sistematizando com jogos matemáticos. Os participantes demonstraram empenho, dedicação e interesse em relação aos truques, jogos e aos algoritmos alternativos. O objetivo do trabalho foi alcançado. Percebemos que houve de fato uma melhoria na aprendizagem dos alunos.

Seja p um número primo e seja uma raiz p - ésima primitiva da unidade. Considere os seguintes elementos i := 1 + + 2 + ... + i-1 para todo 1 i k do anel Z[] onde k = (p-1)/2. Nesta tese nós descrevemos explicitamente um conjunto gerador para o grupo das unidades do anel de grupo integral ZC2p; representado por U(ZC2p); onde C2p representa o grupo cíclico de ordem 2p e p satisfaz as seguintes condições: S := { -1, , u2, ... uk } gera U(Z[]) e U(Zp) = ou U(Zp)2 = e -1 U(Zp); que são verificadas para p = 7; 11; 13; 19; 23; 29; 53; 59; 61 e 67. Com o intuito de estender tais ideias encontramos um conjunto gerador para U(Z(C2p x C2) e U(Z(C2p x C2 x C2) onde p satisfaz as mesmas condições anteriores acrescidas de uma nova hipótese. Finalmente com o auxílio dos resultados anteriores apresentamos um conjunto gerador das unidades centrais do anel de grupo Z(Cp x Q8); onde Q8 representa o grupo dos quatérnios, ou seja, Q8 := <a; b : a4 = 1; a2 = b2; b-1 a b = a-1 >. / Let p be an odd prime integer, be a pth primitive root of unity, Cn be the cyclic group of order n, and U(ZG) the units of the Integral Group Ring ZG: Consider ui := 1++2 +: : :+i1 for 2 i p + 1 2 : In our study we describe explicitly the generator set of U(ZC2p); where p is such that S := f1; ; u2; : : : ; up1 2 g generates U(Z[]) and U(Zp) is such that U(Zp) = 2 or U(Zp)2 = 2 and 1 =2 U(Zp)2; which occurs for p = 7; 11; 13; 19; 23; 29; 37; 53; 59; 61, and 67: For another values of p we don\'t know if such conditions hold. In addition, under suitable hypotheses, we extend these ideas and build a generator set of U(Z(C2p C2)) and U(Z(C2p C2 C2)): Besides that, using the previous results, we exhibit a generator set for the central units of the group ring Z(Cp Q8) where Q8 represents the quaternion group.

Cette thèse porte sur des classes d’objets combinatoires, qui modélisent des données en bio-informatique. Nous étudions notamment deux méthodes de mutation des gènes à l’intérieur du génome : la duplication et l’inversion. Nous étudions d’une part le problème de la duplication-miroir complète avec perte aléatoire en termes de permutations à motifs exclus. Nous démontrons que la classe de permutations obtenue avec cette méthode après p duplications à partir de l’identité est la classe de permutations qui évite les permutations alternées de longueur 2p + 1. Nous énumérons également le nombre de duplications nécessaires et suffisantes pour obtenir une permutation quelconque de longueur n à partir de l’identité. Nous proposons également deux algorithmes efficaces permettant de reconstituer deux chemins différents entre l’identité et une permutation déterminée. Nous donnons enfin des résultats connexes sur d’autres classes proches. La restriction de la relation d’ordre < induite par le code de Gray réfléchi à l’ensemble des compositions et des compositions bornées induit de nouveaux codes de Gray pour ces ensembles. La relation d’ordre < restreinte à l’ensemble des compositions bornées d’un intervalle fournit encore un code de Gray. L’ensemble des ncompositions bornées d’un intervalle généralise simultanément l’ensemble produit et l’ensemble des compositions d’un entier et donc la relation < définit de façon unifiée tous ces codes de Gray. Nous réexprimons les codes de Gray de Walsh et Knuth pour les compositions (bornées) d’un entier à l’aide d’une unique relation d’ordre. Alors, le code de Gray deWalsh pour des classes de compositions et de permutations devient une sous-liste de celui de Knuth, lequel est à son tour une sous-liste du code de Gray réfléchi. / This thesis considers classes of combinatorial objects that model data in bioinformatics. We have studied two methods of mutation of genes within the genome : duplication and inversion. At first,we study the problem of the whole mirror duplication-random lossmodel in terms of pattern avoiding permutations. We prove that the class of permutations obtained with this method after p duplications from the identity is the class of permutations avoiding alternating permutations of length 2p + 1.We also enumerate the number of duplications that are necessary and sufficient to obtain any permutation of length n from the identity. We also suggest two efficient algorithms to reconstruct two different paths between the identity and a specified permutation. Finally,we give related results on other classes nearby. The restriction of the order relation < induced by the reflected Gray code for the sets of compositions and bounded compositions gives new Gray codes for these sets. The order relation < restricted to the set of bounded compositions of an interval also yields a Gray code. The set of bounded n-compositions of an interval simultaneously generalizes product set and compositions of an integer, and so < puts under a single roof all theseGray codes.We re-expressWalsh’s and Knuth’sGray codes for (bounded) compositions of an integer in terms of a unique order relation, and so Walsh’s Gray code becomes a sublist of Knuth’s code, which in turn is a sublist of the Reflected Gray Code.

This diploma thesis deals with the number line as a tool to improve pupil's understanding of various mathematical concepts, structures, properties and algorithms. The aim of the thesis is to determine how are lower secondary pupils able to use the number line and to connect findings with the content of their textbooks. The thesis consists of a theoretical and a practical part. The theoretical part introduces the topic of number line, deals with the problems used for testing in the Czech Republic and presents several researches on this topic. The theoretical part, together with the analysis of current state of the number line usage in teaching, serves as a basis for an experimental investigation that is covered in the practical part. After a pilot testing with 19 pupils, the main testing was conducted in 7 classes of two Prague primary schools. The total of 156 pupils participated in the main testing and 8 of the pupils were later interviewed about their tests. The results are presented in the practical part. It turned out that there is a correspondence between textbook content and pupils' errors while working with number line and that pupils struggle with representing of fractions on the number line. In the conclusion some recommendation for teachers were put forward concerning the usage of the...

Continuation of the 12th part: Complement and correction of the novel Tietken-Castell-Prime-Algorithm