Acta Mathematica Hungarica

, Volume 158, Issue 1, pp 17–26 | Cite as

On the solutions of a Lebesgue–Nagell type equation

  • S. Bhatter
  • A. HoqueEmail author
  • R. Sharma


We find all positive integer solutions in x, y and n of \({x^{2}+19^{2k+1}=4y^{n}}\) for any non-negative integer k.

Key words and phrases

Diophantine equation Lebesgue–Nagell type equation integer solution Lucas sequence primitive divisor 

Mathematics Subject Classification

primary 11D61 secondary 11D41 11R29 


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The authors are grateful to Prof. Kalyan Chakraborty for his careful reading, helpful comments and suggestions. R. Sharma would like to thank to Harish-Chandra Research Institute (HRI) and Malaviya National Institute of Technology, Jaipur for providing sufficient facility to prepare this manuscript. The authors would like to thank the referee whose suggestions and comments help to improve the manuscript.


  1. 1.
    Arif, S.A., Muriefah, F.S.A.: The diophantine equation \(x^2+3^m=y^n\). Internat. J. Math. Math. Sci. 21, 619–620 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arif, S.A., Muriefah, F.S.A.: On the Diophantine equation \(x^2+q^{2k+1}=y^n\). J. Number Theory 95, 95–100 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bennett, M.A., Skinner, C.M.: Ternary Diophantine equations via Galois representations and modular forms. Canad. J. Math. 56, 23–54 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bérczes, A., Pink, I.: On the Diophantine equation \(x^2+p^{2k}=y^n\). Arch. Math. 91, 505–517 (2008)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bérczes, A., Pink, I.: On the Diophantine equation \(x^2+d^{2\ell +1}=y^n\). Glasg. Math. J. 54, 415–428 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Y. Bilu, G. Hanrot and P.  M.  Voutier, Existence of primitive divisors of Lucas and Lehmer numbers (with an appendix by M. Mignotte), J. Reine Angew. Math., 539 (2001), 75–122Google Scholar
  7. 7.
    Y. Bugeaud, M. Mignotte and S. Siksek, Classical and modular approaches to exponential Diophantine equations. II. The Lebesgue–Nagell equation, Compos. Math., 142 (2006), 31–62Google Scholar
  8. 8.
    Chakraborty, K., Hoque, A., Kishi, Y., Pandey, P.P.: Divisibility of the class numbers of imaginary quadratic fields. J. Number Theory 185, 339–348 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    K. Chakraborty and A. Hoque, Exponents of class groups of certain imaginary quadratic fields, preprint, arXiv:1801.00392v1
  10. 10.
    Cohn, J.H.E.: The Diophantine equation \(x^2+C=y^n\). II, Acta Arith. 109, 205–206 (2003)CrossRefzbMATHGoogle Scholar
  11. 11.
    Cohn, J.H.E.: The Diophantine equation \(x^{2}+C = y^{n}\). Acta Arith. 55, 367–381 (1993)CrossRefzbMATHGoogle Scholar
  12. 12.
    Hoque, A., Saikia, H.K.: On the divisibility of class numbers of quadratic fields and the solvability of diophantine equations. SeMA J. 73, 213–217 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Le, M.: On the Diophantine equations \(d_{1}x^{2}+2^{2m}d_{2}=y^{n}\) and \(d_{1}x^{2}+d_{2}=4y^{n}\). Proc. Amer. Math. Soc. 118, 67–70 (1993)MathSciNetGoogle Scholar
  14. 14.
    Le, M.: On the number of solutions of the Diophantine equation \(x^{2}+D = p^{n}\), C. R. Acad. Sci. Paris Sér. A 317, 135–138 (1993)Google Scholar
  15. 15.
    Le, M.: A note on the generalised Ramanujan-Nagell equation. J. Number Theory 50, 193–201 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lebesgue, V.A.: Sur l'impossibilité en nombres entiers de l'équation \(x^m =y^2+1\). Nouvelles Annales des Math. 9, 178–181 (1850)Google Scholar
  17. 17.
    Luca, F.: On a Diophantine equation. Bull. Austral. Math. Soc. 61, 241–246 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    F. Luca, Sz. Tengely and A. Togbe, On the Diophantine equation \(x^2 + C = 4y^n\), Ann. Sci. Math. Quebec, 33 (2009), 171–184Google Scholar
  19. 19.
    Mignotte, M., de Weger, B.M.M.: On the equations \(x^{2}+74 =y^{5}\) and \(x^{2}+ 86 = y^{5}\). Glasgow Math. J. 38, 77–85 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    F. S. A. Muriefah, F. Luca, S.  Siksek and Sz. Tengely, On the Diophantine equation \(x^2 + C = 2y^n\), Int. J. Number Theory, 5 (2009), 1117–1128Google Scholar
  21. 21.
    N. Saradha and A. Srinivasan, Solutions of some generalized Ramanujan-Nagell equations, Indag. Math. (N.S.), 17 (2006), 103–114Google Scholar
  22. 22.
    Sz. Tengely, On the Diophantine equation \(x^2 +q^{2m} = 2y^p\), Acta Arith., 127 (2007), 71–86Google Scholar
  23. 23.
    Zhu, H., Le, M.: On some generalized Lebesque-Nagell equations. J. Number Theory 131, 458–469 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Malaviya National Institute of Technology JaipurJaipurIndia
  2. 2.Harish-Chandra Research InstituteAllahabadIndia

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