Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Solving Thue equations
without the full unit group

Author: Guillaume Hanrot
Journal: Math. Comp. 69 (2000), 395-405
MSC (1991): Primary 11Y50; Secondary 11B37
Published electronically: May 19, 1999
MathSciNet review: 1651759
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The main problem when solving a Thue equation is the computation of the unit group of a certain number field. In this paper we show that the knowledge of a subgroup of finite index is actually sufficient. Two examples linked with the primitive divisor problem for Lucas and Lehmer sequences are given.

References [Enhancements On Off] (What's this?)

  • 1. A. BAKER, G. WÜSTHOLZ, Logarithmic forms and group varieties, J. Reine Angew. Math. 442 (1993), 19-62. MR 94i:11050
  • 2. M. BENNETT, B.M.M. DE WEGER, On the Diophantine equation $|ax^n-by^n|=1$, Math. Comp. 67 (1998), 413-438. MR 98c:11024
  • 3. YU. BILU, G. HANROT, Solving Thue equations of high degree, J. Number Th. 60 (1996), 373-392. MR 97k:11040
  • 4. YU. BILU, G. HANROT, Thue equations with composite fields, Acta Arith., to appear.
  • 5. J. BUCHMANN, A subexponential algorithm for the determination of class groups and regulators of algebraic number fields, Séminaire de Théorie des Nombres de Paris 1988-89, Progr. Math., vol. 94, Birkhaüser, 27-39. MR 92g:11125
  • 6. H. COHEN ``A Course in Computational Algebraic Number Theory'', Graduate Texts in Math., Vol. 138, Springer, 1993. MR 94i:11105
  • 7. H. COHEN, F. DIAZ Y DIAZ, M. OLIVIER, Subexponential algorithms for class group and unit computations, J. Symb. Comp. 24 (1997), 433-441. MR 98m:11138
  • 8. A. COSTA, E. FRIEDMAN, Ratios of regulators in totally real extensions of number fields, J. Number Th. 37 (1991), 288-297. MR 92j:11138
  • 9. J. HAFNER, K. MCCURLEY, A rigorous subexponential algorithm for computation of class groups, J. Amer. Math. Soc. 2 (1989), 837-850. MR 91f:11090
  • 10. G. HANROT, ``Résolution effective d'équations diophantiennes : algorithmes et applications'', Thèse, Université Bordeaux 1, 1997.
  • 11. A. LENSTRA, H.W. LENSTRA, JR., L. LOVÁSZ, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 515-534. MR 84a:12002
  • 12. F.J. VAN DER LINDEN, Class numbers computations of real abelian number fields, Math. Comp. 39 (1982), 693-707. MR 84e:12005
  • 13. J.M. MASLEY, Class numbers of real cyclic number fields with small conductor, Compositio Math. 37 (1978), 297-319. MR 80e:12005
  • 14. M. MIGNOTTE, B.M.M. DE WEGER, On the Diophantine equations $x^2 + 74 = y^5$ and $x^2+76=y^5$, Glasgow Math. J. 38 (1996), 77-85. MR 97b:11044
  • 15. A. PETH\H{O}, Computational methods for the resolution of diophantine equations, in R.A. Mollin (ed.), Number Theory: Proc. First Conf. Can. Number Th. Assoc., Banff, 1988, de Gruyter, 1990, 477-492. MR 92c:11152
  • 16. M. POHST, K. WILDANGER, Tables of unit groups and class groups of quintic fields and a regulator bound, Math. Comp. 67 (1998), 361-367. MR 98d:11163
  • 17. A. SCHINZEL, Primitive divisors of the expression $A^n-B^n$ in algebraic number fields, J. Reine Angew. Math. 268/269 (1974), 27-33. MR 49:8961
  • 18. C.L. STEWART, On divisors of Fermat, Fibonacci, Lucas and Lehmer sequences, Proc. London Math. Soc. (3) 35 (1977), 425-447. MR 58:10694
  • 19. N.P. SMART, The solution of triangularly connected decomposable form equation, Math. Comp. 64 (1995), 819-840. MR 95f:11115
  • 20. R. J. STROEKER, B.M.M. DE WEGER, On elliptic Diophantine equations that defy Thue's method: The case of the Ochoa curve, Exper. Math. 2 (1994), 209-220. MR 96c:11033
  • 21. N. TZANAKIS, B.M.M. DE WEGER, On the practical solution of the Thue Equation, J. Number Th. 31 (1989), 99-132. MR 90c:11018
  • 22. N. TZANAKIS, B.M.M. DE WEGER, How to explicitly solve a Thue-Mahler equation, Compositio Math. 84 (1992), 223-288; 89 (1993), 241-242. MR 93k:11025; MR 95a:11030
  • 23. P. VOUTIER, Primitive divisors of Lucas and Lehmer sequences, Math. Comp. 64 (1995), 869-888. MR 95f:11022
  • 24. P. VOUTIER, Primitive divisors of Lucas and Lehmer sequences, II, J. Th. Nombres Bordeaux 8 (1996), 251-275. MR 98h:11037

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (1991): 11Y50, 11B37

Retrieve articles in all journals with MSC (1991): 11Y50, 11B37

Additional Information

Guillaume Hanrot
Affiliation: Algorithmique Arithmétique Expérimentale, UPRES A CNRS 5465, Université Bordeaux 1, 351, Cours de la Libération, F-33405 Talence Cedex, FRANCE
Address at time of publication: LORIA, 615, rue du Jardin Botanique, B.P. 101, F-54600 Villers-lès-Nancy, FRANCE

Keywords: Diophantine equations, Thue equation, linear recurrence sequences, Lucas sequences, Lehmer sequences, fundamental units
Received by editor(s): April 7, 1997
Received by editor(s) in revised form: March 31, 1998
Published electronically: May 19, 1999
Additional Notes: Partially supported by GDR AMI and GDR Théorie Analytique des Nombres.
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society