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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Complete solutions of a family
of quartic Thue and index form equations


Authors: Maurice Mignotte, Attila Pethö and Ralf Roth
Journal: Math. Comp. 65 (1996), 341-354
MSC (1991): Primary 11D25, 11D57, 11R16, 11Y50
MathSciNet review: 1316596
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Abstract | References | Similar Articles | Additional Information

Abstract: Continuing the recent work of the second author, we prove that the diophantine equation

\begin{displaymath}f_a(x,y)=x^4-ax^3 y-x^2 y^2+axy^3+y^4=1 \end{displaymath}

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 $\gamma\in\mathbb Z[\alpha]$ with $\mathbb Z[\gamma]=\mathbb Z[\alpha]$.


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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
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

DOI: http://dx.doi.org/10.1090/S0025-5718-96-00662-X
PII: S 0025-5718(96)00662-X
Keywords: Thue equation, index form equation, linear forms in the logarithms of algebraic numbers, distributed computation
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
Article copyright: © Copyright 1996 American Mathematical Society