Frações contínuas e aproximações diofantinas

Abstract

The continuous fractions are a method that, based on the principle of recurrence, generates a sequence of rational numbers that progressively approximate a real number. This sequence is known as the convergent sequence. The present work aims to analyze and present theories related to this method, such as the Dirichlet’s Theorem, which establishes an upper limit for the approximation error, and the Hurwitz-Markov Theorem, which, together with the constant √5, optimizes this error, among others. Finally, we try to understand the characterization of periodic continuous fractions and their relation with quadratic equations with integer coefficients, with emphasis on the Pell equation and its resolution by this method.