# Introduction to Diophantine Approximation

Open access

## 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].

[1] Alan Baker. A Concise Introduction to the Theory of Numbers. Cambridge University Press, 1984.

[2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.

[3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.

[4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.

[5] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.

[6] Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.

[7] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.

[8] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.

[9] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.

[10] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.

[11] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.

[12] G.H. Hardy and E.M. Wright. An Introduction to the Theory of Numbers. Oxford University Press, 1980.

[13] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1): 35-40, 1990.

[14] Peter Jaeger. Elementary introduction to stochastic finance in discrete time. Formalized Mathematics, 20(1):1-5, 2012. doi:10.2478/v10037-012-0001-5.

[15] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.

[16] Andrzej Kondracki. The Chinese Remainder Theorem. Formalized Mathematics, 6(4): 573-577, 1997.

[17] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.

[18] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.

[19] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.

[20] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990.

[21] Bo Li, Yan Zhang, and Artur Korniłowicz. Simple continued fractions and their convergents. Formalized Mathematics, 14(3):71-78, 2006. doi:10.2478/v10037-006-0009-9.

[22] Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265-268, 1997.

[23] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.

[24] Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.

[25] Christoph Schwarzweller. Proth numbers. Formalized Mathematics, 22(2):111-118, 2014. doi:10.2478/forma-2014-0013.

[26] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.

[27] Andrzej Trybulec and Czesław Bylinski. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.

[28] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.

[29] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.

[30] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.

[31] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

[32] Bo Zhang, Hiroshi Yamazaki, and Yatsuka Nakamura. Set sequences and monotone class. Formalized Mathematics, 13(4):435-441, 2005.

# Formalized Mathematics

## (a computer assisted approach)

### Journal Information

SCImago Journal Rank (SJR) 2017: 0.119
Source Normalized Impact per Paper (SNIP) 2017: 0.237

Target Group

researchers in the fields of formal methods and computer-checked mathematics

### Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 147 147 9