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

E. P. Davlet'yarova ; A. A. Zhukova ; A. V. Shutov

Čebyševskij sbornik, Tome 17 (2016) no. 2, pp. 88-112

Résumé

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.

Détail

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