Experimental Mathematics

On elliptic Diophantine equations that defy Thue's method: the case of the Ochoa curve

Benjamin M. M. de Weger and Roel J. Stroeker

Abstract

The purpose of this paper is to show that elliptic diophantine equations cannot always be solved--in the most practical sense--by the Thue approach, that is, by solving each of the finitely many corresponding Thue equations of degree 4. After a brief general discussion, which is necessarily of a heuristic nature, to substantiate our claim, we consider the elliptic equation associated with the Ochoa curve. An explicit computational explanation as to the reasons for the failure of the Thue approach in this case is followed by a complete solution of the standard Weierstraß equation of this elliptic curve by a method which makes use of a recent lower bound for linear forms in elliptic logarithms.

Article information

Source
Experiment. Math., Volume 3, Issue 3 (1994), 209-220.

Dates
First available in Project Euclid: 24 March 2003

Permanent link to this document
https://projecteuclid.org/euclid.em/1048515872

Mathematical Reviews number (MathSciNet)
MR1329370

Zentralblatt MATH identifier
0824.11012

Subjects
Primary: 11D25: Cubic and quartic equations
Secondary: 11G05: Elliptic curves over global fields [See also 14H52] 11Y50: Computer solution of Diophantine equations

Keywords
Diophantine equation Thue equation elliptic curve

Citation

Stroeker, Roel J.; de Weger, Benjamin M. M. On elliptic Diophantine equations that defy Thue's method: the case of the Ochoa curve. Experiment. Math. 3 (1994), no. 3, 209--220. https://projecteuclid.org/euclid.em/1048515872


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