Complete solutions of a family of quartic Thue and index form equations
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- by Maurice Mignotte, Attila Pethö and Ralf Roth PDF
- Math. Comp. 65 (1996), 341-354 Request permission
Abstract:
Continuing the recent work of the second author, we prove that the diophantine equation \[ f_a(x,y)=x^4-ax^3 y-x^2 y^2+axy^3+y^4=1 \] for $|a|\ge 3$ has exactly 12 solutions except when $|a|=4$, when it has 16 solutions. If $\alpha =\alpha (a)$ denotes one of the zeros of $f_a(x,1)$, then for $|a|\ge 4$ we also find all $\gamma \in \Bbb Z[\alpha ]$ with $\Bbb Z[\gamma ]=\Bbb Z[\alpha ]$.References
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Additional Information
- Maurice Mignotte
- Affiliation: Université Louis Pasteur, 7 rue René Descartes, 67084 Strasbourg Cedex, France
- Email: mignotte@math.u-strasbourg.fr
- Attila Pethö
- Affiliation: Department of Computer Science, Kossuth Lajos University, P.O. Box 12, H-4010 Debrecen, Hungary
- MR Author ID: 189083
- Email: pethoe@peugeot.dote.hu
- Ralf Roth
- Affiliation: FB-14 Informatik, Universität des Saarlandes, Postfach 151150, D-66041 Saar- brücken, Germany
- Email: roth@cs.uni-sb.de
- Received by editor(s): March 3, 1992
- Received by editor(s) in revised form: February 25, 1993, September 27, 1993, March 15, 1994, and June 2, 1994
- Additional Notes: Research partly done while the second author was a visiting professor at the Fachbereich 14 - Informatik, Universität des Saarlandes
- © Copyright 1996 American Mathematical Society
- Journal: Math. Comp. 65 (1996), 341-354
- MSC (1991): Primary 11D25, 11D57, 11R16, 11Y50
- DOI: https://doi.org/10.1090/S0025-5718-96-00662-X
- MathSciNet review: 1316596