Curvas elípticas e o caso n = 4 da conjectura de Euler

• Ano de defesa: 2014
• Autor(a) principal: Oliveira, Nathália Moraes de
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• Idioma: por
• Instituição de defesa: Universidade Federal de Uberlândia; BR; Programa de Pós-graduação em Matemática; Ciências Exatas e da Terra; UFU
• Palavras-chave em Português: Números de Euler; Conjectura de Euler; Curvas elípticas; Reciprocidade quadrática; Euler s conjecture; Elliptic curves; Quadratic reciprocity; CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA
• Link de acesso: https://repositorio.ufu.br/handle/123456789/16807; https://doi.org/10.14393/ufu.di.2014.88

From Fermat s last theorem we know that the equation X3 +Y 3 = Z3 does not have nontrivial integral solutions. In 1769 Euler conjectured that this result may be generalized increasing the powers and the number of variables. In this work we give a counterexample to Euler s conjecture in the case of n = 4 showing that the equation A4 + B4 + C4 = D4 has nontrivial integral solutions. To do that we study plane algebraic curves, elliptic curves and use results from number theory, especially those on quadratic reciprocity. The quadratic reciprocity is the key factor in the choice of a particular elliptic curve, such that a solution in that elliptic curve becomes a nontrivial solution of the Euler s equation with n = 4. Finally, the arithmetic of the elliptic curves allows us to find infinite integral solutions for A4 + B4 + C4 = D4.