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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Hurwitz’ theorem
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by J. H. E. Cohn
Proc. Amer. Math. Soc. 38 (1973), 436
DOI: https://doi.org/10.1090/S0002-9939-1973-0313195-9

Abstract:

If $[{a_0},{a_1},{a_2}, \cdots ]$ is the continued fraction for a real number $x$, and ${p_n}/{q_n}$ the $n$th convergent, define ${\theta _n} = {q_n}|{p_n} - x{q_n}|$. Hurwitz’ Theorem asserts that ${\phi _n} = \min \{ {\theta _{n - 1}},{\theta _n},{\theta _{n + 1}}\} < {5^{ - 1/2}}$ whenever ${\phi _n}$ is defined. It is the object of this note to provide a simple proof of this fact.
References
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Bibliographic Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 38 (1973), 436
  • MSC: Primary 10F05
  • DOI: https://doi.org/10.1090/S0002-9939-1973-0313195-9
  • MathSciNet review: 0313195