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Mathematics of Computation

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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

N. Tzanakis
Affiliation: Department of Mathematics, University of Crete, Iraklion, Greece

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