Powers of Two in Generalized Fibonacci Sequences

The $k-$generalized Fibonacci sequence $\big(F_{n}^{(k)}\big)_{n}$ resembles the Fibonacci sequence in that it starts with $0,\ldots,0,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. In this paper, we are interested in finding powers of two that appear in $k-$generalized Fibonacci sequences; i.e., we study the Diophantine equation $F_n^{(k)}=2^m$ in positive integers $n,k,m$ with $k\geq 2$.

Powers of Two in Generalized Fibonacci Sequences Potencias de dos en sucesiones generalizadas de Fibonacci JHON J. BRAVO1, FLORIAN LUCA2

1Universidad del Cauca, Popayán, Colombia. Email: jbravo@unicauca.edu.co 
2Universidad Nacional Autónoma de México, Morelia, México. Email:fluca@matmor.unam.mx 

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Abstract

The k-generalized Fibonacci sequence \big(Fn(k)\big)n resembles the Fibonacci sequence in that it starts with 0,…,0,1 (k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we are interested in finding powers of two that appear in k-generalized Fibonacci sequences; i.e., we study the Diophantine equationFn(k)=2m in positive integers n,k,m with k≥ 2.

Key words: Fibonacci numbers, Lower bounds for nonzero linear forms in logarithms of algebraic numbers.

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2000 Mathematics Subject Classification: 11B39, 11J86.

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Resumen

La sucesión k-generalizada de Fibonacci \big(Fn(k)\big)n se asemeja a la sucesión de Fibonacci, pues comienza con 0,…,0,1 (k términos) y a partir de ahí, cada término de la sucesión es la suma de los k precedentes. El interés en este artículo es encontrar potencias de dos que aparecen en sucesiones k-generalizadas de Fibonacci; es decir, se estudia la ecuación Diofántica Fn(k)=2m en enteros positivos n,k,m con k≥ 2.

Palabras clave: Números de Fibonacci, cotas inferiores para formas lineales en logaritmos de números algebraicos.

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Texto completo disponible en PDF

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References

[1] J. J. Bravo and F. Luca, 'k-Generalized Fibonacci Numbers with only one Distinct Digit', Preprint, (2011).

[2] Y. Bugeaud, M. Mignotte, and S. Siksek, 'Classical and Modular Approaches to Exponential Diophantine Equations. I. Fibonacci and Lucas Perfect Powers', Ann. of Math. 163, 3 (2006), 969-1018.

[3] R. D. Carmichael, 'On the Numerical Factors of the Arithmetic Forms αn\pm βn', The Annals of Mathematics15, 1/4 (1913), 30-70.

[4] G. P. Dresden, 'A Simplified Binet Formula for k-Generalized Fibonacci Numbers', Preprint, arXiv:0905.0304v1, (2009).

[5] A. Dujella and A. Pethö, 'A Generalization of a Theorem of Baker and Davenport', Quart. J. Math. Oxford 49, 3 (1998), 291-306.

[6] E. M. Matveev, 'An Explicit Lower Bound for a Homogeneous Rational Linear Form in the Logarithms of Algebraic Numbers', Izv. Math. 64, 6 (2000), 1217-1269.

[7] D. A. Wolfram, 'Solving Generalized Fibonacci Recurrences', The Fibonacci Quarterly 36, 2 (1998), 129-145.

(Recibido en noviembre de 2011. Aceptado en marzo de 2012)

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Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCMv46n1a05, 
AUTHOR = {Bravo, Jhon J. and Luca, Florian}, 
TITLE = {{Powers of Two in Generalized Fibonacci Sequences}}, 
JOURNAL = {Revista Colombiana de Matemáticas}, 
YEAR = {2012}, 
volume = {46}, 
number = {1}, 
pages = {67--79} 
}