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