Computing all integer solutions of a genus 1 equation

Stroeker, R.; Tzanakis, N.

doi:10.1090/S0025-5718-03-01497-2

Computing all integer solutions of a genus 1 equation
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by R. J. Stroeker and N. Tzanakis PDF

Math. Comp. 72 (2003), 1917-1933 Request permission

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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