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
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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 ${\theta _n} = {q_n}|{p_n} - x{q_n}|$. 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
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