On Bachet's diophantine equation \(x^ 3=y^ 2+k\)
type: Item
instance of: scholarly article
title: On Bachet's diophantine equation \(x^ 3=y^ 2+k\)
zbMATH Open document ID: 0545.10010
DOI: 10.1007/BF01507747
author: Ulrich Felgner
published in: Monatshefte für Mathematik
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178199
review text: Using Eisenstein's law of cubic reciprocity we investigate cases in which \(x^ 3=y^ 2+k\) is unsolvable in the ring of rational integers \({\mathbb{Z}}\). In particular we show, that for all primes \(p\not\equiv \pm 1\quad(mod 9),\) \(p\neq 3\), the equation \(x^ 3=y^ 2+3p\quad(p\pm 9)\) has no solutions in \({\mathbb{Z}}\).
Mathematics Subject Classification ID: 11D25
zbMATH DE Number: 3867421
zbMATH Keywords: cubic diophantine equation
zbMATH Keywords: Mordell equation
zbMATH Keywords: unsolvability in the ring of rational integers
MaRDI profile type: Publication
cites work: Q4090395
cites work: Q4106142
cites work: Some Equations y 2 = x 3 −k without Integer Solutions
cites work: Zur Theorie der abstrakten elliptischen Funktionenkörper III. Die Struktur des Meromorphismenrings. Die Riemannsche Vermutung.
cites work: Q5582375
cites work: Numbers of solutions of equations in finite fields
OpenAlex ID: W2029316051
description: scientific article; zbMATH DE number 3867421