O reticulado inteiro e algumas aplicações

• Ano de defesa: 2020
• Autor(a) principal: Euzébio, George da Costa
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• Idioma: por
• Palavras-chave em Português: Reticulado inteiro.; Frações de Farey.; Círculos de Ford.; Teorema de Hurwitz.; Whole lattice.; Farey fractions.; Ford circles.; Hurwitz's theorem.
• Link de acesso: http://www.repositorio.ufc.br/handle/riufc/50517

The present work has as main objective to present the Hurwitz Theorem about the approximation of irrational numbers through rational numbers using a constant associated with the root value of 5 for the approximations. For that, some knowledge about Number Theory, such as divisibility and modular congruences, as well as some theorems about prime and integer numbers that can be written as a sum of two squares, will be recalled. Subsequently, the lattice of the whole numbers and some of the properties of their points will be presented, as well as some geometric theorems about such points. In a last part, the Farey sequence will be developed together with the Ford circles, to present the theorems that guarantee the convergence of continuous fractions and, thus, the proof of Hurwitz's theorem.