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 Chebyshevskii Sb., 2017, Volume 18, Issue 2, Pages 275–278 (Mi cheb558)

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Periodicity and non-periodicity of finite sequences

V. G. Chirskii

Moscow State Pedagogical University

Abstract: Here we study a problem, concerned with generating pseudo-random sequences. The non-periodicity is one of crucial properties of a good pseudo-random sequence. But an infinite non-periodic sequence may have initial segment with improper behavior. For example, the decimal expansion of the Liouvillean number
$$\sum\limits_{n=0}^\infty 10^{-n!}$$
contains only a few digits, equal to one, and all the other are equal to zero.
For practical purposes, therefore, we need to introduce notions of periodicity and sufficient non-periodicity of finite sequences.
The paper treats certain decimal expansions of real numbers and the links between their arithmetic properties and sufficient non-periodicity of these expansions.
Several ways to generate numbers with sufficiently non-periodic expansions are discussed. We overview certain results in this direction and possible ways to develop them further.
We briefly describe problems with polyadic expansions. They are rather convenient since they don't involve division.
The known results are decribed and certain problems formulated.
Bibliography: 11 titles.

Keywords: finite periodicity, arithmetic properties.

DOI: https://doi.org/10.22405/2226-8383-2017-18-2-275-278

Full text: PDF file (580 kB)
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UDC: 517
Accepted:12.06.2017

Citation: V. G. Chirskii, “Periodicity and non-periodicity of finite sequences”, Chebyshevskii Sb., 18:2 (2017), 275–278

Citation in format AMSBIB
\Bibitem{Chi17} \by V.~G.~Chirskii \paper Periodicity and non-periodicity of finite sequences \jour Chebyshevskii Sb. \yr 2017 \vol 18 \issue 2 \pages 275--278 \mathnet{http://mi.mathnet.ru/cheb558} \crossref{https://doi.org/10.22405/2226-8383-2017-18-2-275-278} \elib{http://elibrary.ru/item.asp?id=30042562}