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

 

Computing all integer solutions of a genus 1 equation


Authors: R. J. Stroeker and N. Tzanakis
Journal: Math. Comp. 72 (2003), 1917-1933
MSC (2000): Primary 11D41, 11G05
Published electronically: January 8, 2003
MathSciNet review: 1986812
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Abstract: The elliptic logarithm method has been applied with great success to the problem of computing all integer solutions of equations of degree $3$ and $4$ defining elliptic curves. We extend this method to include any equation $f(u,v)=0$, where $f\in\mathbb{Z}[u,v]$ is irreducible over $\overline{\mathbb{Q}}$, defines a curve of genus $1$, but is otherwise of arbitrary shape and degree. We give a detailed description of the general features of our approach, and conclude with two rather unusual examples corresponding to equations of degree $5$ and degree $9$.


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Additional Information

R. J. Stroeker
Affiliation: Econometric Institute, Erasmus University, P. O. Box 1738, 3000 DR Rotterdam, The Netherlands
Email: stroeker@few.eur.nl

N. Tzanakis
Affiliation: Department of Mathematics, University of Crete, Iraklion, Greece
Email: tzanakis@math.uch.gr

DOI: http://dx.doi.org/10.1090/S0025-5718-03-01497-2
PII: S 0025-5718(03)01497-2
Keywords: Diophantine equation, elliptic curve, elliptic logarithm
Received by editor(s): January 28, 2002
Published electronically: January 8, 2003
Article copyright: © Copyright 2003 American Mathematical Society