Abstract
To each nontotally real cubic extension K of Q and to each generator α of the cubic field K, we attach a family of cubic Thue equations, indexed by the units of K, and we prove that this family of cubic Thue equations has only a finite number of integer solutions, by giving an effective upper bound for these solutions.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
The lower bound follows from looking at the norm!
References
H. Cohen, Advanced topics in computational number theory, in Graduate Texts in Mathematics, vol 193 (Springer, New York, 2000)
C. Levesque, M. Waldschmidt, Familles d’équations de Thue–Mahler n’ayant que des solutions triviales Acta Arith., 155, 117–138 (2012)
T.N. Shorey, R. Tijdeman, Exponential Diophantine equations, in Cambridge Tracts in Mathematics, vol 87 (Cambridge University Press, Cambridge, 1986)
M. Waldschmidt, Diophantine approximation on linear algebraic groups, in Grundlehren der Mathematischen Wissenschaften, vol 326 ( Springer, Berlin, 2000)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this paper
Cite this paper
Levesque, C., Waldschmidt, M. (2013). Families of Cubic Thue Equations with Effective Bounds for the Solutions. In: Borwein, J., Shparlinski, I., Zudilin, W. (eds) Number Theory and Related Fields. Springer Proceedings in Mathematics & Statistics, vol 43. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6642-0_12
Download citation
DOI: https://doi.org/10.1007/978-1-4614-6642-0_12
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-6641-3
Online ISBN: 978-1-4614-6642-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)