On the diophantine equation \(2x^ 3+1=py^ 2\)
type: Item
instance of: scholarly article
title: On the diophantine equation \(2x^ 3+1=py^ 2\)
zbMATH Open document ID: 0588.10017
DOI: 10.1007/BF01171704
published in: Manuscripta Mathematica
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155106
review text: Let \(p\equiv 1 (mod 4)\) be a prime number such that the field \(K={\mathbb{Q}}(\sqrt{-3p})\) has class number 2. Then the diophantine (hyperelliptic) equation (1) \(X^ 6+4=pY^ 2\) has an integral solution [\textit{C. Meyer}, Symp. Math. 15, 365-387 (1975; Zbl 0329.12007), and the reviewer, J. Reine Angew. Math. 339, 27-81 (1983; Zbl 0495.12005)]. If \(p\equiv 5 (mod 8)\) then the equation (1) can be easily solved. In case \(p\equiv 1 (mod 8)\) the field \(K={\mathbb{Q}}(\sqrt{-3p})\) has class number 2 only for \(p=17,41,89\) and solutions of (1) are \((p,| X|,| Y|)=(17,2,2), (41,4,10), (89,10,106).\) Using the p-adic method it is proved in the paper under review that for \(p=17,41,89\) no other solutions exist.
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 11D41
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 3943913
zbMATH Keywords: cubic diophantine equations
zbMATH Keywords: hyperelliptic equation
zbMATH Keywords: class number 2
zbMATH Keywords: p- adic method
author: Nikos Tzanakis
reviewed by: Jannis A. Antoniadis
MaRDI profile type: Publication
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OpenAlex ID: W2317388969
description: scientific article; zbMATH DE number 3943913