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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  6. M. Mignotte, Verification of a conjecture of E. Thomas. J.  Number Theory 44 (1993), 172–177. Zbl0780.11013MR1225951
  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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