scholarly journals Diophantine quadruples with values in k-generalized Fibonacci numbers

2018 ◽  
Vol 68 (4) ◽  
pp. 939-949
Author(s):  
Carlos Alexis Gómez Ruiz ◽  
Florian Luca
Keyword(s):  
Fibonacci Numbers ◽  
Generalized Fibonacci Numbers ◽  
Generalized Fibonacci Sequences ◽  
Diophantine Quadruples ◽  
Fibonacci Sequences ◽  
Positive Integers

AbstractWe consider for integersk≥ 2 thek–generalized Fibonacci sequencesF(k):=$\begin{array}{} (F_n^{(k)})_{n\geq 2-k}, \end{array} $whose firstkterms are 0, …, 0, 1 and each term afterwards is the sum of the precedingkterms. In this paper, we show that there does not exist a quadruple of positive integersa1<a2<a3<a4such thataiaj+ 1 (i≠j) are all members ofF(k).

Representation of integers by k-generalized Fibonacci sequences and applications in cryptography

2021 ◽  
pp. 2150157
Author(s):  
Salim Badidja ◽  
Ahmed Ait Mokhtar ◽  
Özen Özer
Keyword(s):  
Positive Integer ◽  
Fibonacci Numbers ◽  
Generalized Fibonacci Numbers ◽  
Representation Of Integers ◽  
Tribonacci Numbers ◽  
Generalized Fibonacci Sequences ◽  
Fibonacci Sequences ◽  
Positive Integers

The aim of this paper is to construct a relation between tribonacci numbers and generalized tribonacci numbers. Besides, certain conditions are obtained to generalize the representation of a positive integer [Formula: see text] which is determined in [S. Badidja and A. Boudaoud, Representation of positive integers as a sum of distinct tribonacci numbers, J. Math. Statistic. 13 (2017) 57–61] for a [Formula: see text]-generalized Fibonacci numbers [Formula: see text]. Lastly, some applications to cryptography are given by using [Formula: see text].

scholarly journals On prime factors of the sum of two k-Fibonacci numbers

2018 ◽  
Vol 14 (04) ◽  
pp. 1171-1195
Author(s):  
Carlos Alexis Gómez Ruiz ◽  
Florian Luca
Keyword(s):  
Lower Bound ◽  
Prime Factor ◽  
Fibonacci Numbers ◽  
Prime Factors ◽  
Finite Set ◽  
Generalized Fibonacci Sequences ◽  
Fibonacci Sequences ◽  
Positive Integers

We consider for integers [Formula: see text] the [Formula: see text]-generalized Fibonacci sequences [Formula: see text], whose first [Formula: see text] terms are [Formula: see text] and each term afterwards is the sum of the preceding [Formula: see text] terms. We give a lower bound for the largest prime factor of the sum of two terms in [Formula: see text]. As a consequence of our main result, for every fixed finite set of primes [Formula: see text], there are only finitely many positive integers [Formula: see text] and [Formula: see text]-integers which are a non-trivial sum of two [Formula: see text]-Fibonacci numbers, and all these are effectively computable.

On the alternating sums of reciprocal generalized Fibonacci numbers

2021 ◽  
pp. 2250053
Author(s):  
Yücel Türker Ulutaş ◽  
Gökhan Kuzuoğlu
Keyword(s):  
Fibonacci Numbers ◽  
Generalized Fibonacci Numbers ◽  
Alternating Sums ◽  
Greatest Integer Function ◽  
Positive Integers

In this paper, we consider finite alternating sums derived from the generalized Fibonacci numbers [Formula: see text] [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are positive integers with [Formula: see text], [Formula: see text]. Applying the greatest integer function to these sums, we obtain some equalities involving the generalized Fibonacci numbers.

Generalization of the Fibonacci Sequence to n Dimensions

1966 ◽  
Vol 18 ◽  
pp. 332-349 ◽  
Author(s):  
George N. Raney
Keyword(s):  
Free Semigroup ◽  
Jacobi Polynomials ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Natural Generalization ◽  
Unimodular Group ◽  
Natural Extensions ◽  
Generalized Fibonacci Sequences ◽  
Fibonacci Sequences

We introduce certain n X n matrices with integral elements that constitute a free semigroup with identity and generate the n-dimensional unimodular group. In terms of these matrices we define a certain sequence of n-dimensional vectors, which we show is the natural generalization to n dimensions of the Fibonacci sequence. Connections between the generalized Fibonacci sequences and certain Jacobi polynomials are found. The various basic identities concerning the Fibonacci numbers are shown to have natural extensions to n dimensions, and in some cases the proofs are rendered quite brief by the use of known theorems on matrices.

scholarly journals A Quadratic Diophantine Equation Involving Generalized Fibonacci Numbers

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1010 ◽  
Author(s):  
Ana Paula Chaves ◽  
Pavel Trojovský
Keyword(s):  
Diophantine Equation ◽  
Fibonacci Numbers ◽  
Generalized Fibonacci Numbers ◽  
Quadratic Diophantine Equation ◽  
Positive Integers

The sequence of the k-generalized Fibonacci numbers ( F n ( k ) ) n is defined by the recurrence F n ( k ) = ∑ j = 1 k F n − j ( k ) beginning with the k terms 0 , … , 0 , 1 . In this paper, we shall solve the Diophantine equation F n ( k ) = ( F m ( l ) ) 2 + 1 , in positive integers m , n , k and l.

scholarly journals Motzkin’s Maximal Density and Related Chromatic Numbers

2018 ◽  
Vol 13 (1) ◽  
pp. 27-45
Author(s):  
Anshika Srivastava ◽  
Ram Krishna Pandey ◽  
Om Prakash
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Chromatic Numbers ◽  
Distance Graphs ◽  
Maximal Density ◽  
Generalized Fibonacci Numbers ◽  
Generalized Fibonacci Sequence ◽  
Odd Order ◽  
Upper Density ◽  
Positive Integers

Abstract This paper concerns the problem of determining or estimating the maximal upper density of the sets of nonnegative integers S whose elements do not differ by an element of a given set M of positive integers. We find some exact values and some bounds for the maximal density when the elements of M are generalized Fibonacci numbers of odd order. The generalized Fibonacci sequence of order r is a generalization of the well known Fibonacci sequence, where instead of starting with two predetermined terms, we start with r predetermined terms and each term afterwards is the sum of r preceding terms. We also derive some new properties of the generalized Fibonacci sequence of order r. Furthermore, we discuss some related coloring parameters of distance graphs generated by the set M.

SOME ALGEBRAICAL, COMBINATORIAL AND ANALYTICAL PROPERTIES OF PARAGRASSMANN VARIABLES

2005 ◽  
Vol 20 (20n21) ◽  
pp. 4797-4819 ◽  
Author(s):  
MATTHIAS SCHORK
Keyword(s):  
Differential Equations ◽  
Fibonacci Numbers ◽  
Delta Function ◽  
Geometrical Interpretation ◽  
Generalized Fibonacci Numbers ◽  
Analytical Properties ◽  
Witten Index ◽  
Grassmann Variables

Some algebraical, combinatorial and analytical aspects of paragrassmann variables are discussed. In particular, the similarity of the combinatorics involved with those of generalized exclusion statistics (Gentile's intermediate statistics) is stressed. It is shown that the dimensions of the algebras of generalized grassmann variables are related to generalized Fibonacci numbers. On the analytical side, some of the simplest differential equations are discussed and a suitably generalized Berezin integral as well as an associated delta-function are considered. Some remarks concerning a geometrical interpretation of recent results about fractional superconformal transformations involving generalized grassmann variables are given. Finally, a quantity related to the Witten index is discussed.

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