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Linearly periodic continued fractions

Published online by Cambridge University Press:  13 October 2021

Kantaphon Kuhapatanakul  and
Lalitphat Sukruan
Kantaphon Kuhapatanakul
Affiliation:
Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand, e-mail: fscikpkk@ku.ac.th
Lalitphat Sukruan
Affiliation:
Department of Mathematics, Faculty of Science and Technology, Rajamangala University of Technology Suvarnabhumi, Suphanburi, Thailand, e-mail: benjack1001@gmail.com
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Extract

An infinite simple continued fraction representation of a real number α is in the form

$$\eqalign{& {a_0} + {1 \over {{a_1} + {1 \over {{a_2} + {1 \over {{a_3} + {1 \over {}}}}}}}} \cr & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \ddots \cr} $$
where $${a_0}$$ is an integer, and $${a_i}$$ are positive integers for $$i \ge 1$$. This is often written more compactly in one of the following ways:
$${a_0} + {1 \over {{a_1} + }}{1 \over {{a_2} + }}{1 \over {{a_3} + }} \ldots \;{\rm{or}}\;\left[ {{a_0};\;{a_1},\;{a_2},\;{a_3} \ldots } \right]$$
.

Type
Articles
Information
The Mathematical Gazette , Volume 105 , Issue 564 , November 2021 , pp. 442 - 449
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Copyright
© The Mathematical Association 2021

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References

Cohn, H., A short proof of the simple continued fraction expansion of e, Amer. Math. Monthly 113(1) (2006) pp. 57-62.Google Scholar
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Jones, W. B. and Thron, W. J., Continued fractions: analytic theory and applications, Encyclopedia of mathematics and its applications Vol.11, Addison-Wesley Publishing Company (1980).Google Scholar
Laohakosol, V. and Kuhapatanakul, K., The reverse irrationality criteria of Brun and Badea, East-West J. Math., Special volume (2008) pp. 217-234.Google Scholar