On two-parametric quartic families of diophantine problems
type: Item
instance of: scholarly article
title: On two-parametric quartic families of diophantine problems
zbMATH Open document ID: 0926.11016
author: Robert F. Tichy
author: Attila Pethő
published in: Journal of Symbolic Computation
publication date: 1999-12-02T00:00:00Z
review text: The two-parametric Thue equation \[ x(x-y) (x-ay) (x-by)- y^4= \varepsilon, \quad \varepsilon\in \{+1,-1\} \tag{\(*\)} \] is studied for sufficiently large values of the integer parameters \(a,b\). The authors prove that if \[ 10^{2\times 10^{28}}< a+1< b\leq a\Biggl( 1+ \frac{1}{\log^4 a}\Biggr), \] then, the only solutions to \((*)\) are given by \[ (x,y,\varepsilon)= (\pm 1,0,1),\;(0,\pm 1,-1),\;(\pm 1,\pm 1,-1),\;(\pm a,\pm 1,-1),\;(\pm b,\pm 1,-1). \] The authors also prove that if \[ 10^{3\times 10^{18}}\leq a+1< b< a\Biggl(1+ \frac{1}{\log^2 a}\Biggr), \] then the only exceptional units in \(\mathbb{Z} [\alpha]\), where \(\alpha\) is a root of the polynomial \(x(x-1) (x-a)\cdot (x-b)-1\), are the trivial ones. We remind here that a unit \(\eta\) in an order of a number field is called exceptional if \(1-\eta\) is also a unit. In the present situation \(\alpha\) and \(1-\alpha\) are exceptional units. Also, if \(a=2\), then \(\alpha-2\) is also an exceptional unit and, finally, if \(a= b-1\), then \(\alpha-a\) and \(\alpha-a-1\) are also exceptional units. These are the trivial exceptional units mentioned above. As in all papers dealing with explicit solutions of parametric Thue equations, the method of proof is very technical and is based on the theory of linear forms in logarithms and on heavy symbolic computation, which, in the case of the present paper is done with MAPLE.
Mathematics Subject Classification ID: 11D59
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R16
Mathematics Subject Classification ID: 11J86
Mathematics Subject Classification ID: 11Y50
Mathematics Subject Classification ID: 11D25
zbMATH DE Number: 1204261
zbMATH Keywords: Baker's method
zbMATH Keywords: explicit solutions
zbMATH Keywords: two-parametric Thue equation
zbMATH Keywords: exceptional units
zbMATH Keywords: linear forms in logarithms
zbMATH Keywords: MAPLE
reviewed by: Nikos Tzanakis
describes a project that uses: Maple
MaRDI profile type: Publication
full work available at URL: jsco.1998.0205
OpenAlex ID: W2048601908
DOI: 10.1006/JSCO.1998.0205
description: scientific article; zbMATH DE number 1204261