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Chebyshevskii Sb., 2016, Volume 17, Issue 2, Pages 88–112 (Mi cheb481)  

This article is cited in 1 scientific paper (total in 1 paper)

Geometrization of the generalized Fibonacci numeration system with applications to number theory

E. P. Davlet'yarovaab, A. A. Zhukovaab, A. V. Shutovab

a Vladimir State University
b Russian Academy of National Economy and Public Administration under the President of the Russian Federation (Vladimir Branch)

Abstract: Generalized Fibonacci numbers $ \{F ^ {(g)} i \}$ are defined by the recurrence relation
$$ F ^ {(g)} _ {i + 2} = g F ^ {(g)} _ {i + 1} + F ^ {(g)} _ i $$
with the initial conditions $ F ^ {(g)} _ 0 = 1 $, $ F ^ {(g)} _ 1 = g $. These numbers generater representations of natural numbers as a greedy expansions
$$ n = \sum_ {i = 0} ^ {k} \varepsilon_i (n) F ^ {(g)} _ i, $$
with natural conditions on $ \varepsilon_i (n) $. In particular, when $ g = 1 $ we obtain the well-known Fibonacci numeration system. The expansions obtained by $ g> 1 $ are called representations of natural numbers in generalized Fibonacci numeration systems.
This paper is devoted to studying the sets $ \mathbb {F} ^ {(g)} (\varepsilon_0, \ldots, \varepsilon_ {l} ) $, consisting of natural numbers with a fixed end of their representation in the generalized Fibonacci numeration system. The main result is a following geometrization theorem that describe the sets $ \mathbb {F} ^ {(g)} (\varepsilon_0, \ldots, \varepsilon_ {l} ) $ in terms of the fractional parts of the form $ \{n \tau_g \} $, $ \tau_g = \frac {\sqrt {g ^ 2 +4} -g} {2} $. More precisely, for any admissible ending $ (\varepsilon_0, \ldots, \varepsilon_ {l} ) $ there exist effectively computable $ a, b \in \mathbb {Z} $ such that $ n \in \mathbb {F} ^ {(g)} (\varepsilon_0, \ldots, \varepsilon_ {l} ) $ if and only if the fractional part $ \{(n + 1) \tau_g \} $ belongs to the segment $ [\{-a \tau_g \}; \{- b \tau_g \} ] $. Earlier, a similar theorem was proved by authors in the case of classical Fibonacci numeration system.
As an application some analogues of classic number-theoretic problems for the sets $ \mathbb {F} ^ {(g)} (\varepsilon_0, \ldots, \varepsilon_ {l} ) $ are considered. In particular asymptotic formulaes for the quantity of numbers from considered sets belonging to a given arithmetic progression, for the number of primes from considered sets, for the number of representations of a natural number as a sum of a predetermined number of summands from considered sets, and for the number of solutions of Lagrange, Goldbach and Hua Loken problem in the numbers of from considered sets are established.
Bibliography: 33 titles.

Keywords: generalized Fibonacci numeration system, geometrization theorem, distribution in progressions, Goldbach type problem.

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00360_


Full text: PDF file (734 kB)
References: PDF file   HTML file
UDC: 511.43
Received: 05.04.2015
Accepted:10.06.2016

Citation: E. P. Davlet'yarova, A. A. Zhukova, A. V. Shutov, “Geometrization of the generalized Fibonacci numeration system with applications to number theory”, Chebyshevskii Sb., 17:2 (2016), 88–112

Citation in format AMSBIB
\Bibitem{DavZhuShu16}
\by E.~P.~Davlet'yarova, A.~A.~Zhukova, A.~V.~Shutov
\paper Geometrization of the generalized Fibonacci numeration system with applications to number theory
\jour Chebyshevskii Sb.
\yr 2016
\vol 17
\issue 2
\pages 88--112
\mathnet{http://mi.mathnet.ru/cheb481}
\elib{http://elibrary.ru/item.asp?id=26254426}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. A. Zhukova, A. V. Shutov, “Geometrizatsiya sistem schisleniya”, Chebyshevskii sb., 18:4 (2017), 222–245  mathnet  crossref  elib
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