\(S\)-integral solutions to a Weierstrass equation. (English) Zbl 0898.11009

de Weger, Benjamin M. M.

J. Théor. Nombres Bordx. 9, No. 2, 281-301 (1997).

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.

Reviewer: Nigel Smart (Bristol)

MSC:

11D25	Cubic and quartic Diophantine equations
11G05	Elliptic curves over global fields
14G05	Rational points

Keywords:

elliptic curves; cubic diophantine equations; integral points; rational solutions; \(S\)-unit equations; Weierstrass equation

References:

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