Lithuanian Mathematical Journal

, Volume 53, Issue 2, pp 143–148 | Cite as

Repdigit Keith numbers



A Keith number is a positive integer N with the decimal representation a1a2ak such that k ≥ 2 and N appears in the sequence that starts with a1, a2,…, ak and for which each term afterwards is the sum of the k preceding terms. In 2007, Klazar and Luca [M. Klazar and F. Luca, Counting Keith numbers, J. Integer Seq., 10(2):Article 07.2.2, 2007] proved that there are only finitely many Keith numbers with only one distinct digit (so-called repdigits). In this paper, we prove that there are no Keith numbers which are repdigits.


Keith number generalized Fibonacci recurrence repdigits 


11B39 11A63 


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  1. 1.
    J.J. Bravo and F. Luca, On a conjecture about repdigits in k-generalized Fibonacci sequences, Publ. Math., Debrecen. (forthcoming).Google Scholar
  2. 2.
    J.J. Bravo and F. Luca, Powers of two in generalized Fibonacci sequences, Rev. Colomb. Mat., 46(1):67–79, 2012.MathSciNetGoogle Scholar
  3. 3.
    R.P. Brent, On the periods of generalized Fibonacci recurrences, Math. Comput., 63(207):389–401, 1994.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    F.T. Howard and C. Cooper, Some identities for r-Fibonacci numbers, Fibonacci Q., 49(3):231–243, 2011.MathSciNetMATHGoogle Scholar
  5. 5.
    M. Keith, Keith numbers, preprint, available from:
  6. 6.
    M. Keith, Repfigit numbers, J. Recr. Math., 19:41–42, 1987.Google Scholar
  7. 7.
    M. Keith, All repfigit numbers less than 100 billion (1011), J. Recr. Math., 26:181–184, 1994.Google Scholar
  8. 8.
    E. Kilic, The Binet formula, sums and representations of generalized Fibonacci p-numbers, Eur. J. Comb., 29(3):701–711, 2008.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    M. Klazar and F. Luca, Counting Keith numbers, J. Integer Seq., 10(2):Article 07.2.2, 2007.Google Scholar
  10. 10.
    F. Luca, Fibonacci and Lucas numbers with only one distinct digit, Port. Math., 57(2):243–254, 2000.MATHGoogle Scholar
  11. 11.
    D. Marques, On k-generalized Fibonacci numbers with only one distinct digit, Util. Math. (forthcoming).Google Scholar
  12. 12.
    J.B. Muskat, Generalized Fibonacci and Lucas sequences and rootfinding methods, Math. Comput., 61(203):365–372, 1993.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    N.M. Robinson, All known replicating Fibonacci digits less than one thousand billion (1012), J. Recr. Math., 26:188–191, 1994.Google Scholar
  14. 14.
    N. Sloane, The On-Line Encyclopedia of Integer Sequences, available from:

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad del CaucaPopayánColombia
  2. 2.Centro de Ciencias MatemáticasUniversidad Nacional Autónoma de MéxicoMoreliaMexico
  3. 3.Fundación Marcos Moshinsky A.C., Instituto de Ciencias Nucleares de la UNAMMéxico D.F.Mexico
  4. 4.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoMéxico D.F.Mexico

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