<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>k ≥</italic> 1 and denote (<italic>Fk,n</italic>)<italic>n≥</italic>0, the <italic>k</italic>-Fibonacci sequence whose terms satisfy the recurrence relation <italic>Fk,n</italic> = <italic>kFk,n−</italic>1 +<italic>Fk,n−</italic>2, with initial conditions <italic>Fk,</italic>0 = 0 and <italic>Fk,</italic>1 = 1. In the same way, the <italic>k</italic>-Lucas sequence (<italic>Lk,n</italic>)<italic>n≥</italic>0 is defined by satisfying the same recurrence relation with initial values <italic>Lk,</italic>0 = 2 and <italic>Lk,</italic>1 = <italic>k</italic>. These sequences were introduced by Falcon and Plaza, who showed many of their properties, too. In particular, they proved that <italic>Fk,n</italic>+1 + <italic>Fk,n−</italic>1 = <italic>Lk,n</italic>, for all <italic>k ≥</italic> 1 and <italic>n ≥</italic> 0. In this paper, we shall prove that if <italic>k ≥</italic> 1 and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_tmmp-2016-0028_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>F</mi><mrow><mi>k</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mi>s</mi></msubsup><mo>+</mo><msubsup><mi>F</mi><mrow><mi>k</mi><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mi>s</mi></msubsup><mo>∈</mo><msub><mrow><mo>(</mo><mrow><msub><mi>L</mi><mrow><mi>k</mi><mo>,</mo><mi>m</mi></mrow></msub></mrow><mo>)</mo></mrow><mrow><mi>m</mi><mo>≥</mo><mn>1</mn></mrow></msub></math><tex-math>$F_{k,n + 1}^s + F_{k,n - 1}^s \in \left( {L_{k,m} } \right)_{m \ge 1} $</tex-math></alternatives></inline-formula> for infinitely many positive integers <italic>n</italic>, then <italic>s</italic> =1.</abstract>

Let k ≥ 1 and denote (Fk,n)n≥0, the k-Fibonacci sequence whose terms satisfy the recurrence relation Fk,n = kFk,n−1 +Fk,n−2, with initial conditions Fk,0 = 0 and Fk,1 = 1. In the same way, the k-Lucas sequence (Lk,n)n≥0 is defined by satisfying the same recurrence relation with initial values Lk,0 = 2 and Lk,1 = k. These sequences were introduced by Falcon and Plaza, who showed many of their properties, too. In particular, they proved that Fk,n+1 + Fk,n−1 = Lk,n, for all k ≥ 1 and n ≥ 0. In this paper, we shall prove that if k ≥ 1 and Fk,n+1s+Fk,n−1s∈(Lk,m)m≥1$F_{k,n + 1}^s  + F_{k,n - 1}^s  \in \left( {L_{k,m} } \right)_{m \ge 1} $ for infinitely many positive integers n, then s =1.

On the Sum of Powers of Two <italic>k</italic>-Fibonacci Numbers which Belongs to the Sequence of <italic>k</italic>-Lucas Numbers

Departament of Mathematics, Faculty of Natural Sciences, University of Hradec Králové, Hradecká 1285, CZ–500-03 Hradec Králové,

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

On the Sum of Powers of Two -Fibonacci Numbers which Belongs to the Sequence of -Lucas Numbers

Let k ≥ 1 and denote (Fk,n)n≥0, the k-Fibonacci sequence whose terms satisfy the recurrence relation Fk,n = kFk,n−1 +Fk,n−2, with initial conditions Fk,0 = 0...

On the Sum of Powers of Two k-Fibonacci Numbers which Belongs to the Sequence of k-Lucas Numbers

Published Online: Feb 25, 2017

Page range: 41 - 46

Received: Sep 30, 2016

DOI: https://doi.org/10.1515/tmmp-2016-0028

Keywords
-Fibonacci number, -Lucas number, Galois theory, Diophantine equation

© 2016 Pavel Trojovský, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

On the Sum of Powers of Two k-Fibonacci Numbers which Belongs to the Sequence of k-Lucas Numbers

Published Online: Feb 25, 2017

Page range: 41 - 46

Received: Sep 30, 2016

DOI: https://doi.org/10.1515/tmmp-2016-0028

Keywords-Fibonacci number, -Lucas number, Galois theory, Diophantine equation

© 2016 Pavel Trojovský, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
-Fibonacci number, -Lucas number, Galois theory, Diophantine equation