All congruent numbers less than 40000
type: Item
instance of: scholarly article
title: All congruent numbers less than 40000
zbMATH Open document ID: 0922.11023
author: Fidel Nemenzo
published in: Proceedings of the Japan Academy. Series A
publication date: 1999-10-04T00:00:00Z
review text: The author finds the complete list of all rational integers \(0<n<42533\) such that the elliptic curve \(E_n\) with a model \(y^2=x^3-n^2x\) has rank over \({\mathbb Q}\) greater than or equal to \(1\). This condition is also equivalent to the one of \(n\) to be a congruent number. This text improves a former computation done by \textit{H. Wada} and \textit{M. Tairo} [Proc. Japan Acad., Ser. A 70, 154-157 (1994; Zbl 0820.14040)]. The author uses \(2\)-descent to reduce the problem to the solvability of certain diophantine equations, following, for example, \textit{J. H. Silverman} and \textit{J. Tate} in [Rational points on elliptic curves, Springer (1992; Zbl 0752.14034)]. Other problems are discussed concerning the elliptic curves \(E_n\), such as to precisely estimate their rank or to explicitly compute some quotients of their Shafarevich-Tate groups.
reviewed by: Federico Pellarin
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 1192515
zbMATH Keywords: congruent numbers
zbMATH Keywords: elliptic curves
zbMATH Keywords: \(L\)-functions
zbMATH Keywords: Shafarevich-Tate groups
zbMATH Keywords: rank
zbMATH Keywords: cubic diophantine equations
MaRDI profile type: Publication
full work available at URL: pjaa.74.29
OpenAlex ID: W1975831374
DOI: 10.3792/PJAA.74.29
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description: scientific article; zbMATH DE number 1192515