Fermat's `primitive solutions' and some arithmetic of elliptic curves
type: Item
instance of: scholarly article
title: Fermat's `primitive solutions' and some arithmetic of elliptic curves
zbMATH Open document ID: 0787.11022
DOI: 10.1016/0019-3577(93)90041-V
author: Jaap Top
published in: Indagationes Mathematicae. New Series
publication date: 1993-12-06T00:00:00Z
review text: Already Fermat studied elliptic curves \(E: y^ 2 = ax^ 4 + bx^ 3 + cx^ 2 + dx + e\) and discovered a method to find rational points on such curves: starting from ``primitive'' points, Fermat described how to construct infinitely many ``derived points''. These derived points turned out to belong to the subgroup generated by the primitive points of the Mordell-Weil group of \(E(\mathbb{Q})\). In particular, Fermat discovered six primitive points of \(E_ 1: y^ 2 = x^ 4 + 4x^ 3 + 10x^ 2 +20x + 1\) and Nagell posed in 1929 the question whether these points are independent in the Mordell-Weil group of \(E_ 1(\mathbb{Q})\). By using the presently known theory for elliptic curves and available software, the author shows (unconditionally) that the Mordell-Weil group of \(E_ 1(\mathbb{Q})\) has rank 2 and that it is generated by two of the primitive points discovered by Fermat.
reviewed by: Jan-Hendrik Evertse
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 469233
zbMATH Keywords: Fermat
zbMATH Keywords: elliptic curves
zbMATH Keywords: Mordell-Weil group
zbMATH Keywords: primitive points
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 469233