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.


Full text (PDF) (free access)

DOI: 10.3336/gm.51.2.02


References:

  1. J. J. Bravo and F. Luca, Powers of two in generalized Fibonacci sequences, Rev. Colombiana Mat. 46 (2012), 67-79.
    MathSciNet    

  2. J. J. Bravo and F. Luca, On a conjecture about repdigits in k-generalized Fibonacci sequences, Publ. Math. Debrecen 82 (2013), 623-639.
    MathSciNet     CrossRef

  3. J. J. Bravo and F. Luca, Coincidences in generalized Fibonacci recurrences, J. Number Theory 133 (2013), 2121-2137.
    MathSciNet     CrossRef

  4. J. J. Bravo and F. Luca, On the largest prime factor of the k-Fibonacci numbers, Int. J. Number Theory 9 (2013), 1351-1366.
    MathSciNet     CrossRef

  5. J. J. Bravo, C. A. Gómez and F. Luca, Powers of two as sums of two k-Fibonacci numbers, Miskolc Math. Notes 17 (2016), 85-100.
    MathSciNet    

  6. C. Cooper and F. T. Howard, Some identities for r-Fibonacci numbers, Fibonacci Quart. 49 (2011), 231-242.
    MathSciNet    

  7. G. P. Dresden and Zhaohui Du, A simplified Binet formula for k-generalized Fibonacci numbers, J. Integer Seq. 17 (2014), Article 14.4.7.
    MathSciNet    

  8. A. Dujella and A. Pethö, A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser. (2) 49 (1998), 291-306.
    MathSciNet     CrossRef

  9. C. A. Gómez and F. Luca, On the largest prime factor of the ratio of two generalized Fibonacci numbers, J. Number Theory 152 (2015), 182-203.
    MathSciNet     CrossRef

  10. L. K. Hua and Y. Wang, Applications of number theory to numerical analysis, Translated from Chinese. Springer-Verlag, Berlin-New York; Kexue Chubanshe (Science Press), Beijing, 1981.
    MathSciNet    

  11. F. Luca, Fibonacci and Lucas numbers with only one distinct digit, Portugal. Math. 57 (2000), 243-254.
    MathSciNet    

  12. D. Marques, The proof of a conjecture concerning the intersection of k-generalized Fibonacci sequences, Bull. Braz. Math. Soc. (N.S.) 44 (2013), 455-468.
    MathSciNet     CrossRef

  13. D. Marques, On k-generalized Fibonacci numbers with only one distinct digit, Util. Math. 98 (2015), 23-31.
    MathSciNet    

  14. E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers, II, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), 125-180; translation in Izv. Math. 64 (2000), 1217-1269.
    MathSciNet     CrossRef

  15. W. L. McDaniel, Pronic Fibonacci numbers, Fibonacci Quart. 36 (1998), 56-59.
    MathSciNet    

  16. T. D. Noe and J. V. Post, Primes in Fibonacci n-step and Lucas n-step sequences, J. Integer Seq. 8 (2005), Article 05.4.4.
    MathSciNet    

  17. D. A. Wolfram, Solving generalized Fibonacci recurrences, Fibonacci Quart. 36 (1998), 129-145.
    MathSciNet    

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