On a family of cubics
type: Item
instance of: scholarly article
title: On a family of cubics
zbMATH Open document ID: 0734.11025
DOI: 10.1016/0022-314X(91)90032-7
author: Nikos Tzanakis
author: Maurice Mignotte
published in: Journal of Number Theory
publication date: 1991-01-01T00:00:00Z
review text: This is one more step towards the ``practical solution'' of Thue equations invented by the second author and \textit{B. M. M. de Weger} [J. Number Theory 31, 99-132 (1989; Zbl 0657.10014)]. After solving single equations, now the method is able to solve families of equations. The first one was the ``simplest'' cubic family \[ X^ 3-(n-1)X^ 2Y- (n+2)XY^ 2-Y^ 3=\pm 1, \] which was completely solved [see \textit{E. Thomas}, ibid 34, 235-250 (1990; Zbl 0697.10011)] and the first author [Verification of a conjecture of Thomas, Preprint, Strasbourg]. The second example, investigated in this paper, is the family of cubic equations \[ (*)\quad f(X,Y):\;X^ 3-nX^ 2Y-(n+1)XY^ 2-Y^ 3=1. \] A new difficulty arose now, because the cubic extension given by one root of \(f(X,1)=0\) isn't Galois any more. Nevertheless, the authors are able to show that the only solutions of (*) for \(n>3.67\times 10^{32}\) are \[ (1,0),\quad (0,-1),\quad (1,-1),\quad (-n-1,-1),\quad (1,-n). \]
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 11R16
zbMATH DE Number: 4214114
zbMATH Keywords: cubic!diophantine equations
zbMATH Keywords: practical solution
zbMATH Keywords: Thue equations
zbMATH Keywords: family of cubic equations
reviewed by: Bernd Richter
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4214114