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On schizophrenic patterns in $ b$-ary expansions of some irrational numbers

Author: László Tóth
Journal: Proc. Amer. Math. Soc. 148 (2020), 461-469
MSC (2010): Primary 11A63; Secondary 11B37
Published electronically: November 13, 2019
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Abstract: In this paper we study the $ b$-ary expansions of the square roots of the function defined by the recurrence $ f_b(n)=b f_b(n-1)+n$ with initial value $ f(0)=0$ taken at odd positive integers $ n$, of which the special case $ b=10$ is often referred to as the ``schizophrenic'' or ``mock-rational'' numbers. Defined by Darling in $ 2004$ and studied in more detail by Brown in $ 2009$, these irrational numbers have the peculiarity of containing long strings of repeating digits within their decimal expansion. The main contribution of this paper is the extension of schizophrenic numbers to all integer bases $ b\geq 2$ by formally defining the schizophrenic pattern present in the $ b$-ary expansion of these numbers and the study of the lengths of the non-repeating and repeating digit sequences that appear within.

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László Tóth
Affiliation: Rue des Tanneurs 7, L-6790 Grevenmacher, Grand Duchy of Luxembourg

Keywords: Irrational number, $b$-ary expansion, schizophrenic number
Received by editor(s): May 7, 2019
Received by editor(s) in revised form: May 13, 2019
Published electronically: November 13, 2019
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2019 American Mathematical Society