Glasnik Matematicki, Vol. 50, No. 1 (2015), 35-41.

A GENERALIZATION OF A PROBLEM OF MORDELL

Bo He, Ákos Pintér, Alain Togbé and Nóra Varga

Institute of Mathematics, Aba Normal University, Wenchuan, Sichuan 623000, P. R. China
e-mail: bhe@live.cn

Institute of Mathematics, MTA-DE Research Group "Equations, Functions and Curves", Hungarian Academy of Sciences and University of Debrecen, P. O. Box 12, H-4010 Debrecen, Hungary
e-mail: apinter@science.unideb.hu

Department of Mathematics, Purdue University North Central, 1401 S. U.S. 421, Westville, IN 46391, USA
e-mail: atogbe@pnc.edu

Institute of Mathematics, MTA-DE Research Group "Equations, Functions and Curves", Hungarian Academy of Sciences and University of Debrecen, P. O. Box 12, H-4010 Debrecen, Hungary
e-mail: nvarga@science.unideb.hu


Abstract.   In this paper, we use polygonal and pyramidal numbers Polxm and Pyrxm to extend a problem of Mordell. Then we prove that if m≥ 3,n≥ 3 with (m,n)≠ (50,3), (50,6), all the solutions x and y to the related equation verify max(x,y)< C, where C is an effectively computable constant depending only on m and n.

2010 Mathematics Subject Classification.   11D41, 11J86, 11B39, 11D61.

Key words and phrases.   Diophantine equation, binomial coefficients, polygonal numbers, pyramidal numbers.


Full text (PDF) (free access)

DOI: 10.3336/gm.50.1.04


References:

  1. A. Baker, Bounds for the solutions of the hyperelliptic equation, Math. Proc. Cambridge Philos. Soc. 65 (1969), 439-444.
    MathSciNet     CrossRef

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

  3. A. Bremner, An equation of Mordell, Math. Comp. 29 (1975), 925-928.
    MathSciNet     CrossRef

  4. B. Brindza, On S-integral solutions of the equation ym = f(x), Acta Math. Hungar. 44 (1984), 133-139.
    MathSciNet     CrossRef

  5. B. Brindza, Á. Pintér and S. Turjányi, On equal values of pyramidal and polygonal numbers, Indag. Math. (N.S.) 9 (1998), 183-185.
    MathSciNet     CrossRef

  6. Y. Bugeaud, Bounds for the solutions of superelliptic equations, Compos. Math. 107 (1997), 187-219.
    MathSciNet     CrossRef

  7. J. Cremona and D. Rusin, Efficient solution of rational conics, Math. Comp. 72 (2003), 1417-1441.
    MathSciNet     CrossRef

  8. E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012.
    MathSciNet    

  9. L. E. Dickson, History of the theory of numbers. Vol. II: Diophantine analysis, Chelsea Publishing Co., New York, 1966.
    MathSciNet    

  10. M. Kaneko and K. Tachibana, When is a polygonal pyramid number again polygonal? Rocky Mountain J. Math., 32 (2002), 149-165.
    MathSciNet     CrossRef

  11. L. Hajdu, Á. Pintér, Sz. Tengely and N. Varga, Equal values of figurate numbers, J. Number Theory, 137 (2014), 130-141.
    MathSciNet     CrossRef

  12. W. Ljunggren, A diophantine problem, J. London Math. Soc. (2), 3 (1971), 385-391.
    MathSciNet     CrossRef

  13. L. J. Mordell, Diophantine equations, Pure and Applied Mathematics 30, Academic Press, London-New York, 1969.
    MathSciNet    

  14. The PARI Group. PARI/GP version 2.7.0, 2014. Bordeaux, available from http://pari.math.u-bordeaux.fr.

  15. Á. Pintér and N. Varga, Resolution of a nontrivial Diophantine equation without reduction methods, Publ. Math. Debrecen 79 (2011), 605-610.
    MathSciNet     CrossRef

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