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Zbl 0898.11009
de Weger, Benjamin M.M.
(Weger, Benjamin M. M. de)
$S$-integral solutions to a Weierstrass equation. (English)
[J] J. Théor. Nombres Bordx. 9, No.2, 281-301 (1997). ISSN 1246-7405

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. \par 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.
[Nigel Smart (Bristol)]

MSC 2000:: *11D25 Cubic and quartic diophantine equations
11G05 Elliptic curves over global fields
14G05 Rationality questions, rational points

Keywords: elliptic curves; cubic diophantine equations; integral points; rational solutions; $S$-unit equations; Weierstrass equation

Cited in: Zbl 1065.11014

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