On the application of Skolem's p-adic method to the solution of Thue equations
type: Item
instance of: scholarly article
title: On the application of Skolem's p-adic method to the solution of Thue equations
zbMATH Open document ID: 0674.10012
DOI: 10.1016/0022-314X(88)90098-4
author: Roelof J. Stroeker
author: Nikos Tzanakis
published in: Journal of Number Theory
publication date: 1988-01-01T00:00:00Z
review text: The authors show by rather ingenious methods that the only integer solutions of the diophantine equation \[ (*)\quad x^ 4-4x^ 2y^ 2+y^ 4=-47 \] are \((x,y)=(\pm 2,\pm 3)\) and \((\pm 3,\pm 2)\). This seems to be the first case where a totally real binary quartic has been successfully attempted by methods of algebraic number theory. The authors use a trick of Ljunggren - working in a number field M of degree 8 over \({\mathbb{Q}}\)- to reduce (*) to a p-adic system to which Skolem's method is successfully applied with prime \(p=71\). The numerical hardest job is to find all ``exceptable'' units in M for which task they give a powerful reduction algorithm [see also the second author and \textit{B. M. M. de Weger}, ibid. 31, No.1, 99-132 (1989; Zbl 0657.10014)].
Mathematics Subject Classification ID: 11D25
zbMATH DE Number: 4103171
zbMATH Keywords: Skolem's p-adic method
zbMATH Keywords: Thue equations
zbMATH Keywords: totally real binary quartic
reviewed by: Bernd Richter
MaRDI profile type: Publication
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full work available at URL: 0022-314x(88)90098-4
OpenAlex ID: W2031338305
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description: scientific article; zbMATH DE number 4103171