On the \(abc\) conjecture. II.
type: Item
instance of: scholarly article
title: On the \(abc\) conjecture. II.
zbMATH Open document ID: 1036.11032
DOI: 10.1215/S0012-7094-01-10815-6
author: C. L. Stewart
author: Kunrui Yu
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
review text: The \(abc\) conjecture of Masser and Oesterlé asserts that for all \(\varepsilon>0\) there is a constant \(C\) such that for all nonzero \(a,b,c\in\mathbb Z\) with \(a+b+c=0\) and \((a,b,c)=1\), the inequality \(H(a,b,c)\leq CG^{1+\varepsilon}\) holds, where \(H(a,b,c)=\max\{| a| ,| b| ,| c| \}\) and \(G=G(abc)\) is the product of all primes dividing \(abc\) (without multiplicity). This paper is a sequel to the earlier paper by the authors [\textit{C. L. Stewart} and \textit{K. Yu}, Math. Ann. 291, 225--230 (1991; Zbl 0761.11030)]. It sharpens the bound of that paper, obtaining a bound \(H(a,b,c)<\exp(CG^{1/3}(\log G)^3)\), for an effectively computable constant \(C\). Moreover, for \(n\in\mathbb Z\setminus\{-1,0,1\}\) let \(P(n)\) be the largest prime factor of \(n\), and also let \(P(\pm1)=1\). This paper also shows that there is an effectively computable constant \(C\) such that for all nonzero \(a,b,c\in\mathbb Z\) with \(a+b+c=0\), \((a,b,c)=1\), and \(H(a,b,c)>2\), the inequality \(H(a,b,c)<\exp(P'G^{C\log\log\log G^{*}/\log\log G})\) holds, where \(G^{*}=\max\{G,16\}\) and \(P'=\max\{P(a),P(b),P(c)\}\). As was the case with the authors' earlier paper, the theorems of this paper are proved using A. Baker's theory of logarithmic forms, especially a recent estimate of \textit{K. Yu} [Acta Arith. 89, 337--378 (1999; Zbl 0928.11031)].
Mathematics Subject Classification ID: 11J25
Mathematics Subject Classification ID: 11D75
Mathematics Subject Classification ID: 11J86
zbMATH DE Number: 1820817
zbMATH Keywords: effective result
zbMATH Keywords: \(abc\) conjecture
zbMATH Keywords: Baker's theory
zbMATH Keywords: logarithmic forms
reviewed by: Paul Vojta
MaRDI profile type: Publication
Wikidata QID: Q55968424
cites work: Q3837781
cites work: On the \(abc\) conjecture
cites work: Diophantine approximations and value distribution theory
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cites work: Q3211401
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cites work: p-adic logarithmic forms and group varieties II
description: scientific article; zbMATH DE number 1820817