On transformations of periodic sequences

V. G. Chirskii

Čebyševskij sbornik, Tome 17 (2016) no. 3, pp. 191-196

Résumé

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. Bibliography: 15 titles.

Détail

Keywords: periodic sequences, polyadic integers.