On the equation \(y^2=x(x - 2^m)(x+q - 2^m)\)
type: Item
instance of: scholarly article
title: On the equation \(y^2=x(x - 2^m)(x+q - 2^m)\)
zbMATH Open document ID: 1117.11032
author: Andrzej Dąbrowski
author: Małgorzata Wieczorek
published in: Journal of Number Theory
publication date: 2007-05-10T00:00:00Z
review text: Let \(E_{q,m}\) be the elliptic curve defined by \(y^2=x(x-2^m)(x+q-2^m)\). As a generalization of the twin prime conjecture, one may conjecture that, for a given \(m\), there exist infinitely many primes \(q\) such that \(q-2^m\) is also a prime. Thus, it is plausible that the family \[ \{E_{q,m}\,;\, q, q-2^m \text{ are both odd primes}\} \] consists of infinitely many members. The authors study the Mordell-Weil rank of \(E_{q,m}\), and give fairly complete description of the rank using the standard 2-descent method when both \(q\), \(q-2^{m}\) are primes. They subsequently study various other properties of elliptic curves in this family, including the rank of their quadratic twists by prime numbers.
reviewed by: Masato Kuwata
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 11A41
zbMATH DE Number: 5151590
zbMATH Keywords: Elliptic curves
zbMATH Keywords: Mordell-Weil rank
zbMATH Keywords: 2-descent method
zbMATH Keywords: quadratic twists by prime numbers
describes a project that uses: ComputeL
MaRDI profile type: Publication
full work available at URL: j.jnt.2006.09.005
OpenAlex ID: W1975414132
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DOI: 10.1016/J.JNT.2006.09.005
description: scientific article; zbMATH DE number 5151590