Glasnik Matematicki, Vol. 51, No. 2 (2016), 307-319.

MERSENNE K-FIBONACCI NUMBERS

Jhon J. Bravo and Carlos A. Gómez

Departamento de Matemáticas, Universidad del Cauca, Calle 5 No 4-70, Popayán, Colombia
e-mail: jbravo@unicauca.edu.co

Departamento de Matemáticas, Universidad del Valle, Calle 13 No 100-00, Cali, Colombia
e-mail: carlos.a.gomez@correounivalle.edu.co


Abstract.   For an integer k≥ 2, let (Fn(k))n be the k-Fibonacci sequence which starts with 0,...,0,1 (k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we find all k-Fibonacci numbers which are Mersenne numbers, i.e., k-Fibonacci numbers that are equal to 1 less than a power of 2. As a consequence, for each fixed k, we prove that there is at most one Mersenne prime in (Fn(k))n.

2010 Mathematics Subject Classification.   11B39, 11J86.

Key words and phrases.   Generalized Fibonacci numbers, Mersenne numbers, linear forms in logarithms, reduction method.


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DOI: 10.3336/gm.51.2.02


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