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This article is cited in 4 scientific papers (total in 4 papers)
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 $ \left \{F ^ {(g)} i \right \}$
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)}
\left (\varepsilon_0, \ldots, \varepsilon_ {l} \right) $,
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)} \left (\varepsilon_0, \ldots,
\varepsilon_ {l} \right) $ in terms of the fractional parts of the
form $ \left \{n \tau_g \right \} $, $ \tau_g = \frac {\sqrt {g ^
2 +4} -g} {2} $. More precisely, for any admissible ending $ \left
(\varepsilon_0, \ldots, \varepsilon_ {l} \right) $ there exist
effectively computable $ a, b \in \mathbb {Z} $ such that $ n \in
\mathbb {F} ^ {(g)} \left (\varepsilon_0, \ldots, \varepsilon_ {l}
\right) $ if and only if the fractional part $ \left \{(n + 1)
\tau_g \right \} $ belongs to the segment $ \left [\{-a \tau_g \};
\{- b \tau_g \} \right] $. 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)} \left (\varepsilon_0,
\ldots, \varepsilon_ {l} \right) $ 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.
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
Linking options:
https://www.mathnet.ru/eng/cheb481 https://www.mathnet.ru/eng/cheb/v17/i2/p88
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