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Proceedings of the American Mathematical Society

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Hurwitz' theorem


Author: J. H. E. Cohn
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
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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}\vert{p_n} - x{q_n}\vert$. 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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0313195-9
Keywords: Hurwitz Theorem, continued fraction approximation
Article copyright: © Copyright 1973 American Mathematical Society