O décimo problema de Hilbert e uma aplicação

Abstract

This paper deals with Hilbert's Tenth Problem, whose statement is: Given a Diophantine equation with coefficients in any number of variables, it is possible to elaborate a process that decides, through of a finite number of operations, if the equation has integer solutions. The objective is to demonstrate that it is not It is possible to elaborate such a process, that is, to show that Hilbert's Tenth Problem is insoluble. This job begins with a study on Diophantine Equations, Diophantine Sets and Diophantine Functions, analyzing their properties, followed by a proof of the Number Sequence Theorem. A central role in this study is performed by the Pell Equations, used with the purpose of showing that the exponential function is diophantine. This result, along with the concept of recursive function, allows to show that the function is recursive is equivalent to being diophantine. Finally, we prove the Universality Theorem that is used in demonstration of the main theorem that affirms the insolubility of Hilbert's Tenth Problem and in the last chapter is given an application of this result for the demonstration of Gödel's Incomplete Theorem.