Glasnik Matematicki, Vol. 53, No. 1 (2018), 43-50.

ON THE RAMANUJAN-NAGELL TYPE DIOPHANTINE EQUATION X2+AKN=B

Zhongfeng Zhang and Alain Togbé

School of Mathematics and Statistics, Zhaoqing University, Zhaoqing 526061, China
e-mail: bee2357@163.com

Department of Mathematics, Statistics and Computer Science, Purdue University Northwest, 1401 S. U.S. 421 Westville, IN 46391, USA
e-mail: atogbe@pnw.edu

Abstract.   In this paper, we prove that the Ramanujan-Nagell type Diophantine equation x2+Akn=B has at most three nonnegative integer solutions (x, n) for A=1, 2, 4, k an odd prime and B a positive integer. Therefore, we partially confirm two conjectures of Ulas from [23].

2010 Mathematics Subject Classification.   11D41.

Key words and phrases.   Diophantine equation, Pell equations.

Full text (PDF) (free access)

DOI: 10.3336/gm.53.1.04

References:

  1. F. S. Abu Muriefah and Y. Bugeaud, The Diophantine equation x2+c=yn: a brief overview, Rev. Colombiana Mat. 40 (2006), 31-37.
    MathSciNet     CrossRef

  2. R. Apéry, Sur une équation diophantienne, C. R. Acad. Sci. Paris, Sér. A 251 (1960), 1451-1452.
    MathSciNet    

  3. S. A. Arif and F. S. Abu Muriefah, The Diophantine equation x2+q2k=yn, Arab. J. Sci. Eng. Sect. A Sci. 26 (2001), 53-62.
    MathSciNet    

  4. M. Bai and Z. Zhang, On the Ramanujan-Nagell type equation x2+2n=B, Journal of Southwest China Normal University (Natural Science Edition) 42 (2017), 5-7.

  5. M. Bauer and M. Bennett, Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation, Ramanujan J. 6 (2002), 209-270.
    MathSciNet     CrossRef

  6. F. Beukers, On the generalized Ramanujan-Nagell equation I, Acta Arith. 38 (1980/1981), 389-410.
    MathSciNet     CrossRef

  7. F. Beukers, On the generalized Ramanujan-Nagell equation II, Acta Arith. 39 (1981), 114-123.
    MathSciNet     CrossRef

  8. Y. Bugeaud, M. Mignotte and S. Siksek, Classical and modular approaches to exponential Diophantine equations. II. The Lebesgue-Nagell equation, Compos. Math. 142 (2006), 31-62.
    MathSciNet     CrossRef

  9. Y. Bugeaud and T. N. Shorey, On the number of solutions of the generalized Ramanujan-Nagell equation, J. Reine Angew. Math. 539 (2001), 55-74.
    MathSciNet     CrossRef

  10. J. H. E. Cohn, The Diophantine Equation x2+C=yn, Acta. Arith. 65 (1993), 367-381.
    MathSciNet     CrossRef

  11. E. Goins, F. Luca and A. Togbé On the equation x2+2α ∙ 5β ∙ 13γ = yn, in: ANTS VIII Proceedings, eds. A.J. van der Poorten and A. Stein, Lecture Notes in Computer Science 5011, 2008, 430-442.
    MathSciNet     CrossRef

  12. A .Y. Khintchine, Continued fractions, P. Noordhoff Ltd., Groningen, 1963.
    MathSciNet    

  13. C. Ko, On the Diophantine equation x2= yn +1, xy≠ 0, Sci. Sinica 14 (1965), 457-460.
    MathSciNet    

  14. M. Le, A note on the number of solutions of the generalized Ramanujan-Nagell equation x2 - D = kn, Acta Arith. 78 (1996), 11-18.
    MathSciNet     CrossRef

  15. V. A. Lebesgue, Sur l'impossibilité en nombres entiers de l'équation xm = y2+1 Nouv. Annal. des Math. 9 (1850), 178-181.

  16. M. Mignotte and B. M. M. de Weger, On the Diophantine equations x2+74=y5 and x 2+86=y5, Glasgow Math. J. 38 (1996), 77-85.
    MathSciNet     CrossRef

  17. T. Nagell, Introduction to number theory, Almqvist & Wiksell, Stockholm, 1951.
    MathSciNet    

  18. T. Nagell, The Diophantine Equation x2+7=2n, Ark. Math. 4 (191), 185-187.
    MathSciNet     CrossRef

  19. N. Saradha and A. Srinivasan, Generalized Lebesgue-Ramanujan-Nagell equations, in: Diophantine equations, Tata Inst. Fund. Res., Mumbai, 2008, 207-223.
    MathSciNet    

  20. J. Stiller, The Diophantine equation x2 + 119 = 15 ∙ 2n has exactly six solutions, Rocky Mountain J. Math. 26 (1996), 295-298.
    MathSciNet     CrossRef

  21. N. Tzanakis, On the Diophantine equation y2 -D = 2k, J. Number Theory 17 (1983), 144-164.
    MathSciNet     CrossRef

  22. N. Tzanakis and J. Wolfskill, On the diophantine equation y2 = 4qn + 4q + 1, J. Number Theory 23 (1986), 219-237.
    MathSciNet     CrossRef

  23. M. Ulas, Some experiments with Ramanujan-Nagell type Diophantine equations, Glas. Mat. Ser. III 49(69) (2014), 287-302.
    MathSciNet     CrossRef

  24. Z. Zhang and A. Togbé, On two Diophantine equations of Ramanujan-Nagell type, Glas. Mat. Ser. III 51(71) (2016), 17-22.
    MathSciNet     CrossRef
Glasnik Matematicki Home Page