Prime powers in sums of terms of binary recurrence sequences

  • Eshita Mazumdar
  • S. S. RoutEmail author


Let \((u_{n})_{n \ge 0}\) be a non-degenerate binary recurrence sequence with positive discriminant and p be a fixed prime number. In this paper, we are interested in finding a finiteness result for the solutions of the Diophantine equation \(u_{n_{1}} + u_{n_{2}} + \cdots + u_{n_{t}} = p^{z}\) with \(n_1> n_2> \cdots > n_t\ge 0\). Moreover, we explicitly find all the powers of three which are sums of three balancing numbers using lower bounds for linear forms in logarithms. Further, we use a variant of the Baker–Davenport reduction method in Diophantine approximation due to Dujella and Pethő.


Balancing numbers Diophantine equations Linear forms in logarithms Reduction method 

Mathematics Subject Classification

Primary 11B39 Secondary 11D45 11J86 



We thank the referee for suggestions which improved the quality of this paper. The first author would like to thank Harish-Chandra Research Institute, Allahabad and Institute of Mathematics & Applications, Bhubaneswar for their warm hospitality during the academic visits.


  1. 1.
    Behera, A., Panda, G.K.: On the square roots of triangular numbers. Fibonacci Q. 37, 98–105 (1999)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bertók, C., Hajdu, L., Pink, I., Rábai, Z.: Linear combinations of prime powers in binary recurrence sequences. Int. J. Number Theory 13(2), 261–271 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bravo, J.J., Luca, F.: On a conjecture about repdigits in \(k\)-generalized Fibonacci sequences. Publ. Math. Debrecen 82, 623–639 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bravo, E.F., Bravo, J.J.: Power of two as sums of three Fibonacci numbers. Lith. Math. J. 55, 301–311 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bravo, J.J., Gómez, C.A., Luca, F.: Powers of two as sums of two \(k\)-Fibonacci numbers. Miskolc Math. Notes 17, 85–100 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bravo, J.J., Luca, F.: Power of two as sums of two Lucas numbers. J. Integer Seq. 17, A.14.8.3 (2014)Google Scholar
  7. 7.
    Bravo, J.J., Luca, F.: Power of two as sums of two Fibonacci numbers. Quaest. Math. 39, 391–400 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bugeaud, Y., Mignotte, M., Siksek, S.: Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers. Ann Math. 163, 969–1018 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cohn, J.H.E.: On square Fibonacci numbers. J. Lond. Math. Soc. 39, 537–540 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cohn, J.H.E.: Perfect Pell powers. Glasg. Math. J. 38, 19–20 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dey, P.K., Rout, S.S.: Diophantine equations concerning balancing and Lucas balancing numbers. Arch. Math. (Basel) 108(1), 29–43 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dujella, A., Pethő, A.: A generalization of a theorem of Baker and Davenport. Q. J. Math. Oxf. Ser. 49, 291–306 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Marques, D.: Powers of two as sum of two generalized Fibonacci numbers, available at arXiv:1409.2704
  14. 14.
    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
  15. 15.
    Pethő, A.: Perfect powers in second order linear recurrences. J. Number Theory 15, 5–13 (1982)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Pethő, A.: Full cubes in the Fibonacci sequence. Publ. Math. Debr. 30, 117–127 (1983)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Pethő, A., de Weger, B.M.M.: Product of prime powers in binary recurrence sequences. I: The hyperbolic case, with an application to the generalized Ramanujan–Nagell equation. Math. Comp. 47, 713–727 (1986)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Pethő, A.: The Pell sequence contains only trivial perfect powers, Coll. Math. Soc. J. Bolyai, 60 sets, Graphs and Numbers, Budapest, pp. 561–568 (1991)Google Scholar
  19. 19.
    Pink, I., Ziegler, V.: Effective resolution of Diophantine equations of the form \(u_{n} + u_{m} = wp_{1}^{z_{1}} \cdots p_{s}^{z_{s}}\). Monatsh Math. 185, 103–131 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Rout, S.S.: Some Generalizations and Properties of Balancing Numbers, Ph.D. Thesis, NIT Rourkela (2015)Google Scholar
  21. 21.
    Shorey, T.N., Stewart, C.L.: On the Diophantine equation \(ax^{2t}+bx^{t}y+cy^{2}=d\) and pure powers in recurrence sequences. Math. Scand. 52, 24–36 (1983)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Smart, N.P.: The Algorithmic Resolution of Diophantine Equations, London Mathematical Society, Student Texts, Vol 41. Cambridge University Press, Cambridge (1998)Google Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology BombayPowai, MumbaiIndia
  2. 2.Center for CombinatoricsNankai UniversityTianjinChina
  3. 3.Harish-Chandra Research InstituteJhunsiIndia
  4. 4.Institute of Mathematics & ApplicationsAndharua, BhubaneswarIndia

Personalised recommendations