Glasnik Matematicki, Vol. 41, No.1 (2006), 1-8.

COMPUTATIONAL EXPERIENCES ON NORM FORM EQUATIONS FORMING ARITHMETIC PROGRESSIONS

A. Bérczes and A. Pethö

Institute of Mathematics, University of Debrecen, Number Theory Research Group, Hungarian Academy of Sciences and University of Debrecen, H-4010 Debrecen, P.O. Box 12, Hungary
e-mail: berczesa@math.klte.hu

Faculty of Informatics, University of Debrecen, H-4010 Debrecen, P.O. Box 12, Hungary
e-mail: pethoe@inf.unideb.hu


Abstract.   In the present paper we solve the equation

NK/Q(x0 + x1α + x2α2 + ... + xn -1αn -1) = 1

in x0, ... , xn -1 in Z, such that x0, ... , xn -1 is an arithmetic progression, where α is a root of the polynomial xn - a, for all integers 2 ≤ a ≤ 100 and n ≥ 3.

2000 Mathematics Subject Classification.   11D57, 11D59, 11B25.

Key words and phrases.   Norm form equation, arithmetic progression, binomial Thue equation.


Full text (PDF) (free access)

DOI: 10.3336/gm.41.1.01


References:

  1. M. A. Bennett, Rational approximation to algebraic numbers of small height: the Diophantine equation | axn - byn | =1, J. Reine Angew. Math. 535 (2001), 1-49.
    MathSciNet     CrossRef

  2. M. A. Bennett, Products of consecutive integers, Bull. London Math. Soc. 36 (2004), 683-694.
    MathSciNet     CrossRef

  3. M. A. Bennett, K. Györy and Á. Pintér, On the Diophantine equation 1k + 2k + ... + xk = yn, Compos. Math. 140 (2004), 1417-1431.
    MathSciNet

  4. M. A. Bennett and C. M. Skinner, Ternary Diophantine equations via Galois representations and modular forms, Canad. J. Math. 56 (2004), 23-54.
    MathSciNet

  5. M. A. Bennett, V. Vatsal and S. Yazdani, Ternary Diophantine equations of signature (p,p,3), Compos. Math. 140 (2004), 399-1416.
    2098394">MathSciNet

  6. A. Bérczes and A. Pethö, On norm form equations with solutions forming arithmetic progressions, Publ. Math. Debrecen 65 (2004), 281-290.
    MathSciNet

  7. Yu. Bilu, G. Hanrot and P. M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers, J. Reine Angew. Math. 539 (2001), 75-122.
    MathSciNet     CrossRef

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

  9. J. Buchmann and A. Pethö, Computation of independent units in number fields by Dirichlet's method, Math. Comp. 52 (1989), 149-159, S1-S14.
    MathSciNet

  10. H. Darmon and L. Merel, Winding quotients and some variants of Fermat's last theorem, J. Reine Angew. Math. 490 (1997), 81-100.
    MathSciNet

  11. K. Györy, I. Pink and Á. Pintér, Power values of polynomials and binomial Thue-Mahler equations, Publ. Math. Debrecen 65 (2004), 341-362.
    MathSciNet

  12. K. Györy and Á. Pintér, Almost perfect powers in products of consecutive integers, Monatsh. Math. 145 (2005), 19-33.
    MathSciNet     CrossRef

  13. G.Hanrot, Solving Thue equations without the full unit group, Math. Comp. 69 (2000), 395-405.
    MathSciNet     CrossRef

  14. G. Hanrot, N. Saradha and T. N. Shorey, Almost perfect powers in consecutive integers, Acta Arith. 99 (2001), 13-25.
    MathSciNet

  15. A. Kraus, Majorations effectives pour l'équation de Fermat généralisée, Canad. J. Math. 49 (1997), 1139-1161.
    MathSciNet

  16. M. Mignotte, A note on the equation axn - byn = c, Acta Arith. 75 (1996), 287-295.
    MathSciNet

  17. L. J. Mordell, Diophantine equations, Academic Press, London, 1969.
    MathSciNet

  18. Á. Pintér, On the power values of power sums, J. Number Theory, to appear.

  19. K. A. Ribet, On the equation ap + 2α bp + cp = 0, Acta Arith. 79 (1997), 7-16.
    MathSciNet

  20. J.-P. Serre, Sur les représentations modulaires de degré 2 de Gal(Q/Q), Duke Math. J. 54 1987), 179-230.
    MathSciNet     CrossRef

  21. T. N. Shorey and R. Tijdeman, Exponential Diophantine equations, Cambridge Univ. Press, Cambridge-New York, 1986.
    MathSciNet

  22. S. Siksek and J. E. Cremona, On the Diophantine equation x2+7=ym, Acta Arith. 109 (2003), 143-149.
    MathSciNet

  23. W. Stein, The Modular Forms Database,
    http://modular.ucsd.edu/Tables (2004).

  24. The PARIGroup, Bordeaux, PARI/GP, version 2.1.5, 2004,
    available from http://pari.math.u-bordeaux.fr/.

  25. A. Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2) 141 (1995), 443-551.
    MathSciNet     CrossRef

Glasnik Matematicki Home Page