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.
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Notes
- 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)
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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
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DOI: https://doi.org/10.1007/978-1-4614-6642-0_12
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