Skip to main content
SpringerLink
Log in

Common factors of shifted Fibonacci numbers

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

For any positive integer n let Fn be the n-th Fibonacci number. Given positive integers a and b, we study the size of the greatest common divisor of Fn + a and Fm + b for varying positive integers m and n.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Y. Bugeaud, P. Corvaja and U. Zannier, An upper bound for the G.C.D. of a n -1 and b n-1, Preprint, 2001.

  2. P. Corvaja, U. Zannier, A lower bound for the height of a rational function at S-unit points, Preprint, 2003.

  3. P. Corvaja and U. Zannier, Diophantine equations with power sums and universal Hilbert sets, Indag.Math.(N.S.) 9 no. 3 (1998), 317–332.

    MATH  MathSciNet  Google Scholar 

  4. P. Corvaja and U. Zannier, Finiteness of integral values for the ratio of two linear recurrences, Invent.math. 149 no. 2 (2002), 431–451.

    Article  MATH  MathSciNet  Google Scholar 

  5. P. Corvaja and U. Zannier, On the greatest prime factor of (ab + 1)(ac + 1), Proc.Amer.Math.Soc. (to appear).

  6. K. GyŐry, A. SÁrkŐzy and C. L. Stewart, On the number of prime factors of integers of the form ab + 1, Acta Arith. 74 no. 4, 1996.

  7. S. H. HernÁndez and F. Luca, On the largest prime factor of (ab + 1)(ac + 1)(bc + 1), Preprint, 2002.

  8. F. Luca, Divisibility properties of binary recurrent sequences, Indag.Math.(N.S.) 12 no. 3 (2001), 353–367.

    MATH  MathSciNet  Google Scholar 

  9. F. Luca, Arithmetic properties of members of binary recurrent sequences, Acta Arith. 109 no. 1 (2003), 81–107.

    Article  MATH  MathSciNet  Google Scholar 

  10. F. Luca, On the greatest common divisor of two Cullen numbers, Abh.Math.Sem.Univ.Hamburg (to appear).

  11. A. PethŐ, On the greatest prime factor and divisibility properties of linear recursive sequences, Indag.Math.(N.S.) 1, (1990), 85–93.

    MathSciNet  Google Scholar 

  12. T. N. Shorey and R. Tijdeman, Exponential Diophantine equations, Cambridge Univ. Press, Cambridge, 1986.

    MATH  Google Scholar 

  13. A. J. van der Porten and P. G. Walsh, A note on Jacobi symbols and continued fractions, Amer.Math.Monthly 106 no. 1 (1999), 52–56.

    Article  MathSciNet  Google Scholar 

  14. K. R. Yu, p-adic logarithmic forms and group of varieties II, Acta Arith. 89 no. 4 (1999), 337–378.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute UNAM, Ap. Postal 61-3 (Xangari), CP, 58 089, Morelia, Michoacán, Mexico

    Santos Hernández & Florian Luca

Authors
  1. Santos Hernández
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Florian Luca
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hernández, S., Luca, F. Common factors of shifted Fibonacci numbers. Periodica Mathematica Hungarica 47, 95–110 (2003). https://doi.org/10.1023/B:MAHU.0000010814.22264.14

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:MAHU.0000010814.22264.14

Search

Navigation