Complete solutions of a family of cubic Thue equations

Alain Togbé[1]

  • [1] Mathematics Department Purdue University North Central 1401 S, U.S. 421 Westville IN 46391 USA

Journal de Théorie des Nombres de Bordeaux (2006)

  • Volume: 18, Issue: 1, page 285-298
  • ISSN: 1246-7405

Abstract

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In this paper, we use Baker’s method, based on linear forms of logarithms, to solve a family of Thue equations associated with a family of number fields of degree 3. We obtain all solutions to the Thue equation Φ n ( x , y ) = x 3 + ( n 8 + 2 n 6 - 3 n 5 + 3 n 4 - 4 n 3 + 5 n 2 - 3 n + 3 ) x 2 y - ( n 3 - 2 ) n 2 x y 2 - y 3 = ± 1 , for n ≥ 0 .

How to cite

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Togbé, Alain. "Complete solutions of a family of cubic Thue equations." Journal de Théorie des Nombres de Bordeaux 18.1 (2006): 285-298. <http://eudml.org/doc/249648>.

@article{Togbé2006,
abstract = {In this paper, we use Baker’s method, based on linear forms of logarithms, to solve a family of Thue equations associated with a family of number fields of degree 3. We obtain all solutions to the Thue equation\begin\{align*\} \Phi \_n(x,y)&= x^3 + (n^8+2n^6-3n^5+3n^4-4n^3+5n^2-3n+3) x^2 y\\ &\quad - (n^3-2)n^2 x y^2 - y^3 = \pm 1, \end\{align*\}for $n\ge 0$.},
affiliation = {Mathematics Department Purdue University North Central 1401 S, U.S. 421 Westville IN 46391 USA},
author = {Togbé, Alain},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Parametrized Thue equation; Baker's method; Linear forms in logarithms},
language = {eng},
number = {1},
pages = {285-298},
publisher = {Université Bordeaux 1},
title = {Complete solutions of a family of cubic Thue equations},
url = {http://eudml.org/doc/249648},
volume = {18},
year = {2006},
}

TY - JOUR
AU - Togbé, Alain
TI - Complete solutions of a family of cubic Thue equations
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2006
PB - Université Bordeaux 1
VL - 18
IS - 1
SP - 285
EP - 298
AB - In this paper, we use Baker’s method, based on linear forms of logarithms, to solve a family of Thue equations associated with a family of number fields of degree 3. We obtain all solutions to the Thue equation\begin{align*} \Phi _n(x,y)&= x^3 + (n^8+2n^6-3n^5+3n^4-4n^3+5n^2-3n+3) x^2 y\\ &\quad - (n^3-2)n^2 x y^2 - y^3 = \pm 1, \end{align*}for $n\ge 0$.
LA - eng
KW - Parametrized Thue equation; Baker's method; Linear forms in logarithms
UR - http://eudml.org/doc/249648
ER -

References

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  7. M. Pohst, H. Zassenhaus, Algorithmic algebraic number theory. Cambridge University Press, Cambridge, 1989. Zbl0685.12001MR1033013
  8. E. Thomas, Complete solutions to a family of cubic Diophantine equations. J. Number Theory 34 (1990), 235–250. Zbl0697.10011MR1042497
  9. A. Thue, Über Annäherungswerte algebraischer Zahlen. J.  reine angew. Math. 135, 284-305. 
  10. A. Togbé, A parametric family of cubic Thue equations. J. Number Theory 107 (2004), 63–79. Zbl1065.11017MR2059950

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