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{{short description|
In [[number theory]], '''Dirichlet's theorem on Diophantine approximation''', also called '''Dirichlet's approximation theorem''', states that for any [[real numbers]] <math> \alpha </math> and <math> N </math>, with <math> 1 \leq N </math>, there exist integers <math> p </math> and <math> q </math> such that <math> 1 \leq q \leq N </math> and
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==Simultaneous version==
The simultaneous version of the Dirichlet's approximation theorem states that given real numbers <math>\alpha_1, \ldots, \alpha_d</math> and a natural number <math>N</math> then there are integers <math>p_1, \ldots, p_d, q\in\Z,1\le q\leq N^d</math> such that <math>\left|\alpha_i-\frac{p_i}q \right| \le \frac1{qN}.</math><ref>Schmidt,
==Method of proof==
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=== Legendre's theorem on continued fractions ===
{{see also|Simple continued fraction}}
In his ''Essai sur la théorie des nombres'' (1798), [[Adrien-Marie Legendre]] derives a necessary and sufficient condition for a rational number to be a convergent of the [[simple continued fraction]] of a given real number.<ref>{{cite book|last=Legendre|first=Adrien-Marie|author-link=Adrien-Marie Legendre|title=Essai sur la théorie des nombres|date=1798|publisher=Duprat|___location=Paris|publication-date=1798|pages=27–29|language=fr}}</ref> A consequence of this criterion, often called '''Legendre's theorem''' within the study of continued fractions, is as follows:<ref>{{cite journal|last1=Barbolosi|first1=Dominique|last2=Jager|first2=Hendrik|date=1994|title=On a theorem of Legendre in the theory of continued fractions|url=https://www.jstor.org/stable/26273940|journal=[[Journal de Théorie des Nombres de Bordeaux]]|volume=6|issue=1|pages=81–94|doi=10.5802/jtnb.106 |jstor=26273940
'''Theorem'''. If ''α'' is a real number and ''p'', ''q'' are positive integers such that <math>\left|\alpha - \frac{p}{q}\right| < \frac{1}{2q^2}</math>, then ''p''/''q'' is a convergent of the continued fraction of ''α''.
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