Complete solutions to a family of cubic Diophantine equations
type: Item
instance of: scholarly article
title: Complete solutions to a family of cubic Diophantine equations
zbMATH Open document ID: 0697.10011
DOI: 10.1016/0022-314X(90)90154-J
author: Emery Thomas
published in: Journal of Number Theory
publication date: 1990-01-01T00:00:00Z
review text: It is proved in this paper that if \(n\geq 1.365\cdot 10^7\), then the Diophantine equation \[ f_n(x,y)=x^3-(n-1)x^2y-(n+2)xy^2-y^3=\pm 1, \tag{1} \] has only the ``trivial'' solutions \((\pm 1,0)\); \((0,\pm 1)\); \((\pm 1,\mp 1)\). The proof follows that standard method, which is used for the derivation of an effective upper bound for the solutions of Thue equations from an A. Baker's type linear form theorem, but incorporates also a new idea. Let \(x,y\in\mathbb Z^2\) be a solution of (1), \(\alpha(n)\) be a zero of \(f_n(x,1)\) and \(\varepsilon(n)\), \(\eta(n)\) be fundamental units of \(\mathbb Z[\alpha (n)]\). Then there exist integers \(u,v\in\mathbb Z\) with \(x-\alpha(n)y=\varepsilon(n)^u\eta (n)^v\). The new idea is that it is possible to show that if \(uv\neq 0\) then \(\min \{| u|,| v| \}>cn \log n\) with a constant \(c>0\).
Mathematics Subject Classification ID: 11D25
zbMATH DE Number: 4142110
zbMATH Keywords: cubic Diophantine equations
reviewed by: Attila Pethő
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4142110