ON TRANSFORMATIONS OF PERIODIC SEQUENCES
https://doi.org/10.22405/2226-8383-2016-17-3-191-196
Abstract
One of essential problems in generating pseudo-random numbers is the problem of periodicity of the resulting numbers. Some generators output periodic sequences. To avoid it several ways are used.
Here we present the following approach: supposed we have some order in the considered set. Let's invent some algorithm which produces disorder in the set. E.g. if we have a periodic sequence of integers, let's construct an irrational number implying the given set. Then the figures of the resulting number form a non-periodic sequence.
Here we can use continued fractions and Lagrange's theorem asserts that the resulting number is irrational.
Another approach is to use series of the form \(\sum_{n=0}^\infty \frac{a_n}{n!}\) with a periodic sequence of integers \(\{a_n\}, a_{n+T}=a_n\) which is irrational.
Here we consider polyadic series \(\sum_{n=0}^\infty a_n n!\) with a periodic sequence of positive integers \(\{a_n\},a_{n+T} = a_n\) and describe some of their properties.
About the Author
V. G. Chirskii
Moscow pedagogical state University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, associate
professor, head of number theory’s chair
References
1. Knuth. D., 1969, ”The art of computer programming”, Addison-Wesley.
2. Khintchin A. Ya., 1964, ”Continued fractions”, Univ. Chicago Press.
3. Lang S., 1966, ”Introduction to Diophantine approximation”, Addison-Wesley.
4. Chirskii V. G., Nesterenko A. Yu., 2015) , O”n an approach to transforming periodic sequences”, Discretnaya matematica, v. 27, no. 4, pp. 150–157 (Russian)
5. Nesterenko Yu. V., 1995, ”Hermite-Pade approximations of generalized hypergeometric functions”, Engl. transl. Russ. Acad. Sci. Sb. Math, 85, pp. 189–219.
6. Chirskii V. G., 2014, ”Arithmetic properties of polyadic series with periodic coefficients”, Doklady Mathematics, v. 90, no. 3, pp. 766–768.
7. Novoselov E. V., 1982, ”Introduction to polyadic calculus”, Petrozavodsk, 112 p. (Russian)
8. Postnikov A. G., 1971, ”Introduction to analytic number theory”, Moscow, Nauka, 416 p. (Russian)
9. Chirskii V. G., 2015, ”Arithmetic properties of polyadic integers”, Tchebyshevskiy sbornik, v. 16, no. 1, pp. 254–264. (Russian)
10. . Chirskii V. G., Matveev V. Yu., 2013, ”On a series of products of terms of an arithmetic progression”, Prepodavatel 21 veka, no. 4, pp. 245–254. (Russian)
11. Matveev V. Yu., 2013, ”On the values of a certain series at points, well approximable by positive integers”, Prepodavatel 21 veka, no. 4, pp. 339–354. (Russian)
12. Matveev V. Yu., 2016, ”On the infinite algebraic independence of certain polyadic numbers”, Proceedings of an international conference ”Matematica I informatica”, Moscow, March 13-17, pp. 125–126. (Russian)
13. Chirskii V. G., 2016, ”On transformations of periodic sequences”, Proceedings of an international conference ”Matematica I informatica”, Moscow, March 13-17, pp. 143–144. (Russian)
14. Luzhina L. M., Makarov Yu. N., 2016, ”A sufficient condition of the algebraic dependence of polyadic series”, Proceedings of an international conference ”Matematica I informatica”, Moscow, March 13-17, pp. 123–124. (Russian)
15. Chirskii V. G., Matveev V. Yu., 2013, ”On certain properties of polyadic expansions”, Chebyshevskiy sbornik, v. 14, no. 2, pp. 163–172. (Russian)
Review
For citations:
Chirskii V.G. ON TRANSFORMATIONS OF PERIODIC SEQUENCES. Chebyshevskii Sbornik. 2016;17(3):191-196. (In Russ.) https://doi.org/10.22405/2226-8383-2016-17-3-191-196
Views:
681
Email this article 