On the \(abc\) conjecture. II.
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)].
scientific article; zbMATH DE number 1820817