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

DOI:
https://doi.org/10.1090/S0025-5718-03-01497-2

Published electronically:
January 8, 2003

MathSciNet review:
1986812

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

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