
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'yarova^{ab}, A. A. Zhukova^{ab}, A. V. Shutov^{ab} ^{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 wellknown 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 numbertheoretic
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.
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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 88112
\mathnet{http://mi.mathnet.ru/cheb481}
\elib{http://elibrary.ru/item.asp?id=26254426}
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This publication is cited in the following articles:

A. A. Zhukova, A. V. Shutov, “Geometrizatsiya sistem schisleniya”, Chebyshevskii sb., 18:4 (2017), 222–245

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