Dissertação (mestrado)—Universidade de Brasília, Departamento de Matemática, 2013. / Submitted by Alaíde Gonçalves dos Santos (alaide@unb.br) on 2013-07-16T14:10:41Z No. of bitstreams: 1 2013_MatheusBernardinideSouza.pdf: 449129 bytes, checksum: 84b41aaa9be182b0ab45d842af511738 (MD5) / Approved for entry into archive by Leandro Silva Borges(leandroborges@bce.unb.br) on 2013-07-16T17:14:02Z (GMT) No. of bitstreams: 1 2013_MatheusBernardinideSouza.pdf: 449129 bytes, checksum: 84b41aaa9be182b0ab45d842af511738 (MD5) / Made available in DSpace on 2013-07-16T17:14:02Z (GMT). No. of bitstreams: 1 2013_MatheusBernardinideSouza.pdf: 449129 bytes, checksum: 84b41aaa9be182b0ab45d842af511738 (MD5) / O objetivo deste trabalho é mostrar algumas técnicas para resolução de equações diofantinas. Métodos algébricos são ferramentas de grande utilidade para a resolução da equação equation x2 + 7 = yn, em que y = 2 ou Y é ímpar. O uso do método hipergeométrico traz um resultado recente (de 2008) no estudo da equação x2 + 7 =2n. m e técnicas algébricas garantem uma condição necessária para que essa última equação tenha solução. _______________________________________________________________________________________ ABSTRACT / The objective of this work is to show some techniques for solving Diophantine equations. Algebraic methods are useful tools for solving the equation x2 + 7 = yn, where y = 2 or y is odd. The use of the hypergeometric method brings a recent result (from 2008) in the study of the equation x2 + 7 = 2n.m and algebraic techniques ensure a necessary condition for the last equation to have a solution.

Srinivasa Ramanujan (1887-1920) was one of the world's greatest mathematical geniuses. He work extensively in a branch of mathematics called "q-series". Around 1913, he found an important formula which now is known as Ramanujan's 1ψ1summation formula. The aim of this thesis is to investigate Ramanujan's 1ψ1summation formula and explore its applications to number theory and combinatorics. First, we consider several classical important results on elliptic functions and then give new proofs of these results using Ramanujan's 1ψ1 summation formula. For example, we will present a number of classical and new solutions for the problem of representing an integer as sums of squares (one of the most celebrated in number theory and combinatorics) in this thesis. This will be done by using q-series and Ramanujan's 1ψ1 summation formula. This in turn will give an insight into how Ramanujan may have proven many of his results, since his own proofs are often unknown, thereby increasing and deepening our understanding of Ramanujan's work.