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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

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1898-9934
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Volume 23, Issue 2 (Jun 2015)

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Introduction to Diophantine Approximation

Yasushige Watase
  • Suginami-ku Matsunoki 6 3-21 Tokyo, Japan
Published Online: 2015-08-13 | DOI: https://doi.org/10.1515/forma-2015-0010

Abstract

In this article we formalize some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals. A typical example is finding an integer solution (x, y) of the inequality |xθ − y| ≤ 1/x, where 0 is a real number. First, we formalize some lemmas about continued fractions. Then we prove that the inequality has infinitely many solutions by continued fractions. Finally, we formalize Dirichlet’s proof (1842) of existence of the solution [12], [1].

MSC: 11A55; 11J68; 03B35

Keywords: irrational number; approximation; continued fraction; rational number; Dirichlet’s proof

MML: identifier: DIOPHAN1; version: 8.1.045.32.1237

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About the article

Received: 2015-04-19

Published Online: 2015-08-13

Published in Print: 2015-06-01



Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, DOI: https://doi.org/10.1515/forma-2015-0010. Export Citation

© by Yasushige Watase. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

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