Complete solutions of a family of cubic Thue equations
Journal de Théorie des Nombres de Bordeaux, Volume 18 (2006) no. 1, pp. 285-298.

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

Φn(x,y)=x3+(n8+2n6-3n5+3n4-4n3+5n2-3n+3)x2y-(n3-2)n2xy2-y3=±1,

for n0.

Dans cet article, nous utilisons la méthode de Baker, basée sur les formes linéaires en logarithmes, pour résoudre une famille d’équations de Thue liée à une famille de corps de nombres de degré 3. Nous obtenons toutes les solutions de l’équation de Thue

Φn(x,y)=x3+(n8+2n6-3n5+3n4-4n3+5n2-3n+3)x2y-(n3-2)n2xy2-y3=±1,

pour n0.

Received:
Published online:
DOI: 10.5802/jtnb.544
Alain Togbé 1

1 Mathematics Department Purdue University North Central 1401 S, U.S. 421 Westville IN 46391 USA
@article{JTNB_2006__18_1_285_0,
     author = {Alain Togb\'e},
     title = {Complete solutions of a family of cubic {Thue} equations},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {285--298},
     publisher = {Universit\'e Bordeaux 1},
     volume = {18},
     number = {1},
     year = {2006},
     doi = {10.5802/jtnb.544},
     zbl = {05070458},
     mrnumber = {2245886},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.544/}
}
TY  - JOUR
TI  - Complete solutions of a family of cubic Thue equations
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2006
DA  - 2006///
SP  - 285
EP  - 298
VL  - 18
IS  - 1
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.544/
UR  - https://zbmath.org/?q=an%3A05070458
UR  - https://www.ams.org/mathscinet-getitem?mr=2245886
UR  - https://doi.org/10.5802/jtnb.544
DO  - 10.5802/jtnb.544
LA  - en
ID  - JTNB_2006__18_1_285_0
ER  - 
%0 Journal Article
%T Complete solutions of a family of cubic Thue equations
%J Journal de Théorie des Nombres de Bordeaux
%D 2006
%P 285-298
%V 18
%N 1
%I Université Bordeaux 1
%U https://doi.org/10.5802/jtnb.544
%R 10.5802/jtnb.544
%G en
%F JTNB_2006__18_1_285_0
Alain Togbé. Complete solutions of a family of cubic Thue equations. Journal de Théorie des Nombres de Bordeaux, Volume 18 (2006) no. 1, pp. 285-298. doi : 10.5802/jtnb.544. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.544/

[1] A. Baker, Contribution to the theory of Diophantine equations. I. On the representation of integers by binary forms. Philos. Trans. Roy. Soc. London Ser. A 263 (1968), 173–191. | MR: 228424 | Zbl: 0157.09702

[2] T. Cusick, Lower bounds for regulators. In “Number Theory, Noordwijkerhout, 1983,” Lecture Notes in Mathematics, Vol. 1068, pp. 63–73, Springer-Verlag, Berlin/New York, 1984. | MR: 756083 | Zbl: 0549.12003

[3] M. Daberkow, C.  Fieker, J.  Kluners, M. E.  Pohst, K,  Roegner, K.  Wildanger, Kant V4. J. Symbolic Comput. 24 (1997) 267-283. | MR: 1484479 | Zbl: 0886.11070

[4] C. Heuberger, A. Togbé, V. Ziegler, Automatic solution of families of Thue equations and an example of degree 8. J. Symbolic Computation 38 (2004), 145–163. | MR: 2093887

[5] Y. Kishi, A family of cyclic cubic polynomials whose roots are systems of fundamental units. J.  Number Theory 102 (2003), 90–106. | MR: 1994474 | Zbl: 1034.11060

[6] M. Mignotte, Verification of a conjecture of E. Thomas. J.  Number Theory 44 (1993), 172–177. | MR: 1225951 | Zbl: 0780.11013

[7] M. Pohst, H. Zassenhaus, Algorithmic algebraic number theory. Cambridge University Press, Cambridge, 1989. | MR: 1033013 | Zbl: 0685.12001

[8] E. Thomas, Complete solutions to a family of cubic Diophantine equations. J. Number Theory 34 (1990), 235–250. | MR: 1042497 | Zbl: 0697.10011

[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. | MR: 2059950 | Zbl: 1065.11017

Cited by Sources: