Demonstrações elementares para a equação de Fermat x^n + y^n = z^n, para n pertencente a {2,3,4,5}

Abstract

This paper presents some elementary demonstrations for some particular cases of Fermat's Last Theorem. We often seek to find whole solutions for a polynomial equation, that is, to solve a diophantine equation. This idea was the basis of what is considered to be the most famous and enduring problem in the history of mathematics, Fermat's Last Theorem. The search for evidence or evidence of this result was crucial in the development of algebraic theory of numbers, allowing to establishment several tools, powerful and sophisticated, very contributory to modern mathematics. One fact is that Fermat's Last Theorem was one of the great mysteries of the history of mathematics and it challenged the most brilliant and determined minds in the world of mathematics. An easy-to understand theorem that many consider impossible. This paper focuses on presenting some elementary demonstrations for Fermat's Last Theorem xn + yn = 2n for cases n ∈ {2, 3, 4, 5}, showing that such an equation does not have non trivial integer solutions for n> 2. For the case n = 5, a partial demonstration is made, showing that if 5 + x, 5 + y, 5+ , with x, y, z integers, then x5+y5=z5.