The Ramanujan Journal

, Volume 46, Issue 1, pp 49–75 | Cite as

Diophantine equations with products of consecutive members of binary recurrences

  • Attila Bérczes
  • Yuri F. Bilu
  • Florian Luca


We prove a finiteness result for the number of solutions of a Diophantine equation of the form \(u_n u_{n+1}\cdots u_{n+k}\pm 1 =\pm u_m^2\), where \(\{ u_n\}_{n\ge 1}\) is a binary recurrent sequence whose characteristic equation has roots which are real quadratic units.


Diophantine equations Binary recurrences Applications of linear forms in logarithms 

Mathematics Subject Classification

11B39 11D61 



We thank the referee for careful reading and detecting a flaw in the initial version. We also thank Karim Belabas and Michael Mossinghoff for helpful suggestions.


  1. 1.
    Beukers, F.: The multiplicity of binary recurrences. Compos. Math. 40, 251267 (1980)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bravo, J.J., Komatsu, T., Luca, F.: On the distance between products of consecutive Fibonacci numbers and powers of Fibonacci numbers. Indag. Math. 24, 181–198 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Guzman, S., Luca, F.: Linear combinations of factorials and \(S\)-units in a binary recurrence sequence. Ann. Math. Québec 38, 169–188 (2014)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Komatsu, T., Luca, F., Tachiya, Y.: On the multiplicative order of \(F_{n+1}/F_n\) modulo \(F_m\). In: Proceedings of the Integers Conference 2011, Integers, vol. 12, p. A8 (2012/2013)Google Scholar
  5. 5.
    Lehmer, D.H.: Factorization of certain cyclotomic functions. Ann. Math. II(34), 461–479 (1933)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Marques, D.: The Fibonacci version of the Brocard–Ramanujan Diophantine equation. Far East J. Math. Sci. 56, 219–224 (2011)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Masser, D.W.: Linear relations on algebraic groups. In: New Advances in Transcendence Theory (Durham. 1986), pp. 248–262. Cambridge University Press, Cambridge (1988)Google Scholar
  8. 8.
    Matveev, E.M.: An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers II. Izv. Math. 64, 1217–1269 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Szalay, L.: Diophantine equations with binary recurrences associated to Brocard–Ramanujan problem. Port. Math. 69, 213–220 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Szikszai, M.: A variant of the Brocard–Ramanujan equation for Lucas sequences (2016, Preprint)Google Scholar
  11. 11.
    Yu, K.: \(p\)-adic logarithmic forms and group varieties II. Acta Arith. 89, 337–378 (1999)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Attila Bérczes
    • 1
  • Yuri F. Bilu
    • 2
  • Florian Luca
    • 3
    • 4
    • 5
  1. 1.Institute of MathematicsUniversity of DebrecenDebrecenHungary
  2. 2.Institut de Mathématiques de BordeauxUniversité de Bordeaux and CNRSTalenceFrance
  3. 3.School of MathematicsUniversity of the WitwatersrandWitsSouth Africa
  4. 4.Max Planck Institute for MathematicsBonnGermany
  5. 5.Department of Mathematics, Faculty of SciencesUniversity of OstravaOstrava 1Czech Republic

Personalised recommendations