Original scientific paper

https://doi.org/10.3336/gm.51.2.02

Mersenne k-Fibonacci numbers

Jhon J. Bravo orcid.org/0000-0001-7772-9260 ; Departamento de Matemáticas, Universidad del Cauca, Calle 5 No 4-70, Popayán, Colombia
Carlos A. Gómez orcid.org/0000-0003-1126-2973 ; Departamento de Matemáticas, Universidad del Valle, Calle 13 No 100-00, Cali, Colombia

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.

Keywords

Generalized Fibonacci numbers; Mersenne numbers; linear forms in logarithms; reduction method

Hrčak ID:

170038

URI

https://hrcak.srce.hr/170038

Publication date:

3.12.2016.

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