Computing integral points on Mordell's elliptic curves
type: Item
instance of: scholarly article
title: Computing integral points on Mordell's elliptic curves
zbMATH Open document ID: 0865.11084
author: J. Gebel
author: Attila Pethő
author: Horst G. Zimmer
published in: Collectanea Mathematica
publication date: 1996-11-21T00:00:00Z
full work available at URL: https://eudml.org/doc/40418
review text: In an earlier paper [Acta Arith. 68, 171-192 (1994; Zbl 0816.11019)] the authors developed an algorithm for computing all integral points on elliptic curves \(E\) over the rationals \(\mathbb{Q}\). This algorithm, based on a method of Lang, Zagier, was implemented in the computer algebra system SIMATH. (A similar method was used by \textit{R. J. Stroeker} and \textit{N. Tzanakis} for computing all integral points on some elliptic curves.) In the present paper we report on an application of our procedure to Mordell's elliptic curves \(E_k: y^2= x^3+ k\): We compute all integral points for \(0\neq k\in \mathbb{Z}\) within the range \(|k|\leq 10 000\). In fact, we determine more generally all \(S\)-integral points on \(E_k\) for the set of primes \(S= \{2, 3, 5, \infty\}\). The computations can be extended to supply all integral points on \(E_k\) within the range \(|k|< 100 000\). The numerical results obtained are of interest, e.g. in view of Hall's conjecture asserting that the \(x\)-coordinate of an integer point on \(E_k\) satisfies the inequality \[ |x|^{1/ 2}< c|k| \] for some constant \(c>0\). Within the range \(|k|\leq 100 000\), the conjecture holds with \(c=5\). An extended version of this paper will appear in Compos. Math.
Mathematics Subject Classification ID: 11Y16
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14Q05
Mathematics Subject Classification ID: 11D25
zbMATH DE Number: 937454
zbMATH Keywords: algorithm
zbMATH Keywords: Mordell's elliptic curves
zbMATH Keywords: integral points
reviewed by: Horst G. Zimmer
describes a project that uses: SIMATH
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 937454