Complete solution of a family of quartic Thue equations
type: Item
instance of: scholarly article
title: Complete solution of a family of quartic Thue equations
zbMATH Open document ID: 0853.11021
DOI: 10.1007/BF02953340
author: Günter Lettl
author: Attila Pethő
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1996-06-19T00:00:00Z
review text: This is the second example of a family of quartic Thue equations, depending on an integer parameter \(a\), which is completely solved for all values of \(a\). The first analogous example is due to \textit{M. Mignotte}, the second named author and \textit{R. Roth} [Math. Comput. 65, 341-354 (1996; see the following review)]. The authors solve completely the quartic Thue equations \[ F_a (x, y)= x^4- ax^3 y- 6x^2 y^2+ axy^3+ y^4= c, \qquad c\in \{\pm 1, \pm 4\} \] for all values of \(a\). Their method is much in the spirit of the aforementioned paper but, certainly, less technical -- due to the nice properties of the number fields defined by a root of \(F_a (X, 1)\) -- and by far less demanding in computer assistance. These features make the present paper more easily readable than that of Mignotte et al, which means that, if one wants to understand the underlying method in both these papers, one should probably start form the one under review. An interesting proposition characterizing the units of the maximal order of the corresponding quartic fields is also proved.
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 11R16
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11Y50
zbMATH DE Number: 878391
zbMATH Keywords: linear forms in logarithms
zbMATH Keywords: maximal order
zbMATH Keywords: quartic fields
zbMATH Keywords: family of quartic Thue equations
zbMATH Keywords: units
reviewed by: Nikos Tzanakis
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02953340
OpenAlex ID: W2049401768
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description: scientific article; zbMATH DE number 878391