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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Approximation by continued fractions

Author: Melvyn B. Nathanson
Journal: Proc. Amer. Math. Soc. 45 (1974), 323-324
MSC: Primary 10F20
MathSciNet review: 0349594
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Abstract: Let $ x$ be a real irrational number whose continued fraction has infinitely many partial quotients not less than $ k$. A simple proof is given that the inequality $ \vert x - p/q\vert < {({k^2} + 4)^{ - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}}{q^{ - 2}}$ has infinitely many rational solutions $ p/q$. The constant $ {({k^2} + 4)^{ - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}}$ is best possible.

References [Enhancements On Off] (What's this?)

  • [1] J. H. E. Cohn, Hurwitz' theorem, Proc. Amer. Math. Soc. 38 (1973), 436. MR 0313195 (47:1750)
  • [2] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 4th ed., Oxford Univ. Press, London, 1960.

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Keywords: Hurwitz' theorem, continued fraction, diophantine approximation
Article copyright: © Copyright 1974 American Mathematical Society

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