Glasnik Matematicki, Vol. 53, No. 2 (2018), 265-273.

TRINOMIALS ax8+bx+c WITH GALOIS GROUPS OF ORDER 1344

Szabolcs Tengely

Institute of Mathematics, University of Debrecen, P.O.Box 12, 4010 Debrecen , Hungary
e-mail: tengely@science.unideb.hu

Abstract.   Bruin and Elkies ([7]) obtained the curve of genus 2 parametrizing trinomials ax8 + bx + c whose Galois group is contained in G1344 = (ℤ/2)3 ⋊ G168. They found some rational points of small height and computed the associated trinomials. They conjecture that the only ℚ-rational points of the hyperelliptic curve

Y2 = 2X6 + 28X5 + 196X4 + 784X3 + 1715X2 + 2058X + 2401

are given by (X, Y ) = (0, ± 49), (-1, ± 38), (-3, ± 32), and (-7, ± 196). In this paper we prove that the above points are the only S-integral points with S={2,3,5,7,11,13,17,19}.

2010 Mathematics Subject Classification.   11G30, 11Y50

Key words and phrases.   Trinomials, hyperelliptic curves, S-integral points

Full text (PDF) (free access)

DOI: 10.3336/gm.53.2.04

References:

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

  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 and B. K. Spearman, Cyclic sextic trinomials x6+Ax+B, Int. J. Number Theory 6 (2010), 161-167.
    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. S. C. Brown, B. K. Spearman and Q. Yang, On the Galois groups of sextic trinomials, JP J. Algebra Number Theory Appl. 18 (2010), 67-77.
    MathSciNet    

  6. S. C. Brown, B. K. Spearman and Q. Yang, On sextic trinomials with Galois group C6, S3 or C3× S3, J. Algebra Appl. 12 (2013), 1250128, 9 pp.
    MathSciNet     CrossRef

  7. N. Bruin and N. D. Elkies, Trinomials ax7+bx+c and ax8+bx+c with Galois groups of order 168 and 8·168, Algorithmic number theory (Sydney, 2002), Lecture Notes in Comput. Sci. 2369, Springer, Berlin, 2002, 172-188.
    MathSciNet     CrossRef

  8. N. Bruin, Chabauty methods and covering techniques applied to generalized Fermat equations, Dissertation, University of Leiden, Leiden, 1999. CWI Tract 133, Stichting Mathematisch Centrum Centrum voor Wiskunde en Informatica, Amsterdam, 2002.
    MathSciNet    

  9. N. Bruin, Chabauty methods using elliptic curves, J. Reine Angew. Math. 562 (2003) 27-49.
    MathSciNet     CrossRef

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

  11. Y. Bugeaud, M. Mignotte, S. Siksek, M. Stoll, and Sz. Tengely, Integral points on hyperelliptic curves, Algebra Number Theory 2 (2008), 859-885.
    MathSciNet     CrossRef

  12. Y. Bugeaud, M. Mignotte, and S. Siksek, Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers, Ann. of Math. (2) 163 (2006), 969-1018.
    MathSciNet     CrossRef

  13. H. R. Gallegos-Ruiz, S-integral points on hyperelliptic curves, PhD thesis, University of Warwick, 2010.

  14. H. R. Gallegos-Ruiz, S-integral points on hyperelliptic curves, Int. J. Number Theory 7 (2011), 803-824.
    MathSciNet     CrossRef

  15. E. Landau, Verallgemeinerung eines Pólyaschen satzes auf algebraische zahlkörper, 1918.

  16. E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II. Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), 125-180.
    MathSciNet     CrossRef

  17. A. Pethö and B. M. M. de Weger, Products of prime powers in binary recurrence sequences. I. The hyperbolic case, with an application to the generalized Ramanujan-Nagell equation, Math. Comp. 47 (1986), 713-727.
    MathSciNet     CrossRef

  18. V. G. Sprindžuk,The arithmetic structure of integer polynomials and class numbers, Analytic number theory, mathematical analysis and their applications (dedicated to I. M. Vinogradov on his 85th birthday), Trudy Mat. Inst. Steklov. 143 (1977), 152-174.
    MathSciNet    

  19. M. Stoll, On the height constant for curves of genus two, Acta Arith. 90 (1999), 183-201.
    MathSciNet     CrossRef

  20. M. Stoll, Implementing 2-descent for Jacobians of hyperelliptic curves, Acta Arith. 98 (2001), 245-277.
    MathSciNet     CrossRef

  21. M. Stoll, On the height constant for curves of genus two. II, Acta Arith. 104 (2002), 165-182.
    MathSciNet     CrossRef

  22. P. M. Voutier,An upper bound for the size of integral solutions to Ym=f(X), J. Number Theory 53 (1995), 247-271.
    MathSciNet     CrossRef
Glasnik Matematicki Home Page