\(S\)-integral solutions to a Weierstrass equation
type: Item
instance of: scholarly article
title: \(S\)-integral solutions to a Weierstrass equation
zbMATH Open document ID: 0898.11009
author: Benjamin M. M. de Weger
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1998-10-27T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93692
full work available at URL: jtnb9-2.html
review text: The author determines the rational solutions (with a power of \(2\) in the denominator) to the equation \[ y^2 = x^3 - 228 x + 848. \] This is an elliptic curve of rank two. There are a variety of standard techniques to solve this problem: For instance using a reduction to Thue-Mahler equations or using elliptic logarithms (and their \(p\)-adic generalizations). However, in this paper the author takes the novel approach of reduction to four term \(S\)-unit equations of a special form. The author attributes this idea to Y. Bilu. The approach in the current paper has a number of advantages. One does not need unproved estimates for linear forms in \(p\)-adic elliptic logarithms, nor does one need to reduce to a set of Thue-Mahler equations, which can lead to huge computational problems. In addition, the paper contains a detailed explanation of the calculations performed to solve the equation.
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14G05
zbMATH DE Number: 1164088
zbMATH Keywords: elliptic curves
zbMATH Keywords: cubic diophantine equations
zbMATH Keywords: integral points
zbMATH Keywords: rational solutions
zbMATH Keywords: \(S\)-unit equations
zbMATH Keywords: Weierstrass equation
MaRDI profile type: Publication
OpenAlex ID: W2137810378
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DOI: 10.5802/JTNB.203
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description: scientific article; zbMATH DE number 1164088