Glasnik Matematicki, Vol. 49, No. 2 (2014), 287-302.

SOME EXPERIMENTS WITH RAMANUJAN-NAGELL TYPE DIOPHANTINE EQUATIONS

Maciej Ulas

Jagiellonian University, Faculty of Mathematics and Computer Science, Institute of Mathematics, Łojasiewicza 6, 30 - 348 Kraków, Poland
e-mail: maciej.ulas@uj.edu.pl


Abstract.   Stiller proved that the Diophantine equation x2+119=15 · 2n has exactly six solutions in positive integers. Motivated by this result we are interested in constructions of Diophantine equations of Ramanujan-Nagell type x2=Akn+B with many solutions. Here, A,B (thus A, B are not necessarily positive) and k≥ 2 are given integers. In particular, we prove that for each k there exists an infinite set S containing pairs of integers (A, B) such that for each (A,B) S we have gcd(A,B) is square-free and the Diophantine equation x2=Akn+B has at least four solutions in positive integers. Moreover, we construct several Diophantine equations of the form x2=Akn+B with k>2, each containing five solutions in non-negative integers. We also find new examples of equations x2=A2n+B having six solutions in positive integers, e.g. the following Diophantine equations have exactly six solutions:

x2= 57· 2n+117440512,   n=0 , 14 , 16, 20, 24, 25,

x2= 165· 2n+26404,   n=0 , 5 , 7, 8, 10, 12.

Moreover, based on an extensive numerical calculations we state several conjectures on the number of solutions of certain parametric families of the Diophantine equations of Ramanujan-Nagell type.

2010 Mathematics Subject Classification.   11D41.

Key words and phrases.   Diophantine equation, Ramanujan-Nagell equation.


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DOI: 10.3336/gm.49.2.04


References:

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

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

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

  4. F. Beukers, On the generalized Ramanujan-Nagell equation II, Acta Arith. 39 (1981), 113-123.
    MathSciNet    

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

  6. W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (3-4) (1997), 235-265.
    MathSciNet     CrossRef

  7. P. Bundschuh, On the diophantine equation of Ramanujan-Nagell, Semin. Math. Sci. 12 (1988), 31-40.
    MathSciNet    

  8. A. Dabrowski and M. Ulas, Variations on Brocard-Ramanujan equation, J. Number Theory 133 (2013), 1168-1185.
    MathSciNet     CrossRef

  9. G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349-366.
    MathSciNet     CrossRef

  10. H. Hasse, Über eine Diophantische Gleichung von Ramanujan-Nagell und ihre Verallgemeinerung, Nagoya Math. J. 27 (1966), 77-102.
    MathSciNet     CrossRef

  11. T. Nagell, The Diophantine equation x2+7=2n, Ark. Math. 4 (1961), 185-187.
    MathSciNet     CrossRef

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

  13. J. Stiller, The Diophantine equation x2+119=15· 2n has exactly six solutions, Rocky Mountain J. Math. 26 (1) (1996), 295-298.
    MathSciNet     CrossRef

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

  15. M. Ulas, Some observations on the Diophantine equation y2 = x! + A and related results, Bull. Aust. Math. Soc. 86 (2012), 377-388.
    MathSciNet     CrossRef

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