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 A Diophantine equation related to the sum of powers of two consecutive generalized Fibonacci numbers

2015 ◽  
Vol 156 ◽  
pp. 1-14 ◽  
Author(s):  
Ana Paula Chaves ◽  
Diego Marques
Keyword(s):  
Diophantine Equation ◽  
Fibonacci Numbers ◽  
Generalized Fibonacci Numbers

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].

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.

scholarly journals On the Diophantine Equation z(n) = (2 − 1/k)n Involving the Order of Appearance in the Fibonacci Sequence

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 124 ◽  
Author(s):  
Eva Trojovská
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Positive Integers

Let ( F n ) n ≥ 0 be the sequence of the Fibonacci numbers. The order (or rank) of appearance z ( n ) of a positive integer n is defined as the smallest positive integer m such that n divides F m . In 1975, Sallé proved that z ( n ) ≤ 2 n , for all positive integers n. In this paper, we shall solve the Diophantine equation z ( n ) = ( 2 − 1 / k ) n for positive integers n and k.

The Diophantine equation Fn = P(x)

2020 ◽  
Vol 16 (09) ◽  
pp. 2095-2111
Author(s):  
Szabolcs Tengely ◽  
Maciej Ulas
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Infinitely Many Solutions ◽  
Polynomial Formula ◽  
Positive Integers ◽  
Integral Solutions

We consider equations of the form [Formula: see text], where [Formula: see text] is a polynomial with integral coefficients and [Formula: see text] is the [Formula: see text]th Fibonacci number that is, [Formula: see text] and [Formula: see text] for [Formula: see text] In particular, for each [Formula: see text], we prove the existence of a polynomial [Formula: see text] of degree [Formula: see text] such that the Diophantine equation [Formula: see text] has infinitely many solutions in positive integers [Formula: see text]. Moreover, we present results of our numerical search concerning the existence of even degree polynomials representing many Fibonacci numbers. We also determine all integral solutions [Formula: see text] of the Diophantine equations [Formula: see text] for [Formula: see text] and [Formula: see text].

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).

An exponential Diophantine equation related to the sum of powers of two consecutive k-generalized Fibonacci numbers

2014 ◽  
Vol 137 (2) ◽  
pp. 171-188 ◽  
Author(s):  
Carlos Alexis Gómez Ruiz ◽  
Florian Luca
Keyword(s):  
Diophantine Equation ◽  
Fibonacci Numbers ◽  
Exponential Diophantine Equation ◽  
Generalized Fibonacci Numbers

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.

scholarly journals On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1813
Author(s):  
S. Subburam ◽  
Lewis Nkenyereye ◽  
N. Anbazhagan ◽  
S. Amutha ◽  
M. Kameswari ◽  
...  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Exponential Diophantine Equation ◽  
Research Article ◽  
All Solutions ◽  
Sum Of Products ◽  
Consecutive Integers ◽  
Sums Of Products ◽  
Positive Integers

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

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