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

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

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

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_à

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UDC: 511.43
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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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
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