GEOMETRIZATION OF THE GENERALIZED FIBONACCI NUMERATION SYSTEM WITH APPLICATIONS TO NUMBER THEORY

Abstract

Generalized Fibonacci numbers { F (g) i } are defined by the recurrence relation F (g) i+2 = gF(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 = ∑k i=0 εi(n)F (g) i , with natural conditions on εi(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 F (g) (ε0, . . . , εl), 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 F (g) (ε0, . . . , εl) in terms of the fractional parts of the form {nτg}, τg = √ g 2+4−g 2 . More precisely, for any admissible ending (ε0, . . . , εl) there exist effectively computable a, b ∈ Z such that n ∈ F (g) (ε0, . . . , εl) if and only if the fractional part {(n + 1)τg} belongs to the segment [{−aτg}; {−bτg}]. 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 F (g) (ε0, . . . , εl) 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.