Verification of a conjecture of E. Thomas
type: Item
instance of: scholarly article
title: Verification of a conjecture of E. Thomas
zbMATH Open document ID: 0780.11013
author: Maurice Mignotte
published in: Journal of Number Theory
publication date: 1993-10-18T00:00:00Z
review text: For each integer \(n>3\), the Diophantine equation \(X^3 - (n-1) X^2 - (n+2) XY^2 - Y^3=\pm 1\) in integers \(X\) and \(Y\) has only the six trivial solutions \(\pm (0,1)\), \(\pm (1,0)\), \(\pm(1,-1)\). The ranges \(n\geq 1.365\cdot 10^7\) and \(3<n \leq 10^3\) had been covered by \textit{E. Thomas} [J. Number Theory 34, 235--250 (1990; Zbl 0697.10011)]. In the present paper, the author completes the proof by means of sharp lower bounds for linear forms in two logarithms; this is an example where refined numerical constants for such lower bounds play a fundamental role, because the proof involves a large number of linear forms in two logarithms. This solution of Thomas' conjecture provides the first example of a family of diophantine equations which can be solved by transcendence methods. Further examples have been produced more recently [\textit{E. Thomas}, Solutions to certain families of Thue equations, J. Number Theory 43, No. 3, 319--369 (1993; Zbl 0774.11013)].
reviewed by: Michel Waldschmidt
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 11D59
Mathematics Subject Classification ID: 11J82
Mathematics Subject Classification ID: 11J86
zbMATH DE Number: 431318
zbMATH Keywords: cubic Diophantine equations
zbMATH Keywords: Thue equations
zbMATH Keywords: transcendence methods
zbMATH Keywords: sharp lower bounds for linear forms in two logarithms
zbMATH Keywords: Thomas' conjecture
MaRDI profile type: Publication
full work available at URL: jnth.1993.1043
OpenAlex ID: W1972468051
Wikidata QID: Q123295693
DOI: 10.1006/JNTH.1993.1043
description: scientific article; zbMATH DE number 431318