Dirichlet's approximation theorem: Difference between revisions

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== Related theorems ==
=== Legendre's theorem on continued fractions ===
{{see also|ContinuedSimple 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|last=Barbolosi|first=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|via=JSTOR}}</ref>
 
'''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''.