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The Distribution of Self-Fibonacci Divisors

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Advances in the Theory of Numbers

Part of the book series: Fields Institute Communications ((FIC,volume 77))

Abstract

Consider the positive integers n such that n divides the n-th Fibonacci number, and their counting function A. We prove that

$$\displaystyle{A(x) \leq x^{1-(1/2+o(1))\log \log \log x/\log \log x}.}$$

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References

  1. J.J. Alba González, F. Luca, C. Pomerance, I.E. Shparlinski, On numbers n dividing the nth term of a linear recurrence. Proc. Edinb. Math. Soc. 55.2, 271–289 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  2. P. Erdős, F. Luca, C. Pomerance, On the proportion of numbers coprime to a given integer, in Proceedings of the Anatomy of Integers Conference, Montréal, March 2006, ed. by J.-M. De Koninck, A. Granville, F. Luca. CRM Proceedings and Lecture Notes, vol. 46, pp. 47–64 (2008)

    Google Scholar 

  3. D.M. Gordon, C.Pomerance, The distribution of Lucas and elliptic pseudoprimes. Math. Comput. 57, 825–838 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  4. J.H. Halton, On the divisibility properties of Fibonacci numbers. Fibonacci Q. 4.3, 217–240 (1966)

    MATH  MathSciNet  Google Scholar 

  5. F. Luca, C. Pomerance, Combinatorial Number Theory. Irreducible Radical Extensions and Euler-function Chains (de Gruyter, Berlin, 2007), pp. 351–361

    Google Scholar 

  6. D. Marques, Fixed points of the order of appearance in the Fibonacci sequence. Fibonacci Q. 50.4, 346–351 (2012)

    MATH  MathSciNet  Google Scholar 

  7. C. Pomerance, On the distribution of pseudoprimes. Math. Comput. 37, 587–593 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  8. J.B. Rosser, L. Schoenfeld, Approximate formulas for some functions of prime numbers. Ill. J. Math. 6.1, 64–94 (1962)

    MATH  MathSciNet  Google Scholar 

  9. G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory (Cambridge University Press, Cambridge, 1995)

    Google Scholar 

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Acknowledgements

We are especially grateful to the referee for improving the argument in Sect. 3, to P. Leonetti for helping formulate Theorem 3, and to C. Sanna for pointing out a flaw in an earlier version of Theorem 2. We also thank B. Cloitre, K. Ford, D. Marques, C. Pomerance, G. Tenenbaum, L. Versari.

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Authors and Affiliations

  1. School of Mathematics, University of the Witwatersrand, 2050, Wits, South Africa

    Florian Luca

  2. Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126, Pisa, Italy

    Emanuele Tron

Authors
  1. Florian Luca
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  2. Emanuele Tron
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Corresponding author

Correspondence to Florian Luca .

Editor information

Editors and Affiliations

  1. School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada

    Ayşe Alaca

  2. School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada

    Şaban Alaca

  3. School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada

    Kenneth S. Williams

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Luca, F., Tron, E. (2015). The Distribution of Self-Fibonacci Divisors. In: Alaca, A., Alaca, Ş., Williams, K. (eds) Advances in the Theory of Numbers. Fields Institute Communications, vol 77. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3201-6_6

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