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

Abstract.   For an integer k≥ 2, let (F_n^(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 (F_n^(k))_n.

2010 Mathematics Subject Classification.   11B39, 11J86.

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

DOI: 10.3336/gm.51.2.02

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