Some Diophantine Problems Related to k-Fibonacci Numbers

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F62690094%3A18470%2F20%3A50016982" target="_blank" >RIV/62690094:18470/20:50016982 - isvavai.cz</a>
Result on the web
<a href="https://www.mdpi.com/2227-7390/8/7/1047" target="_blank" >https://www.mdpi.com/2227-7390/8/7/1047</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.3390/math8071047" target="_blank" >10.3390/math8071047</a>

Result language
angličtina
Original language name
Some Diophantine Problems Related to k-Fibonacci Numbers
Original language description
Let k >= 1 be an integer and denote (F-k,F-n) n as the k-Fibonacci sequence whose terms satisfy the recurrence relation F-k,F-n=kF(k,n-1)+F-k,F-n-2, with initial conditions F-k,F-0=0 and F-k,F-1=1. In the same way, the k-Lucas sequence (L-k,L-n)(n) is defined by satisfying the same recursive relation with initial values L-k,L-0=2 and L-k,L-1=k. The sequences(F-k,F-n)(n >= 0) and (L-k,L-n)(n >= 0) were introduced by Falcon and Plaza, who derived many of their properties. In particular, they proved that F-k,n(2)+F-k,n+1(2)=F-k,F-2n+1 and F-k,n+1(2)-F-k,n-1(2)=kF(k,2n), for all k >= 1 and n >= 0. In this paper, we shall prove that if k>1 and F-k,n(s)+F-k,n+1(s) is an element of(F-k,F-m)(m >= 1) for infinitely many positive integers n, then s=2. Similarly, that if F-k,n+1(s)-F-k,n-1(s) is an element of(kF(k,m))(m >= 1) holds for infinitely many positive integers n, then s=1 or s=2. This generalizes a Marques and Togbe result related to the case k=1. Furthermore, we shall solve the Diophantine equations F-k,F-n=L-k,L-m, F-k,F-n=F-n,F-k and L-k,L-n=L-n,L-k.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GA18-00496S" target="_blank" >GA18-00496S: Singular spaces from special holonomy and foliations</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2020
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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