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On the \(abc\) conjecture. (English) Zbl 0761.11030
For an integer \(n\), define \(G(n)\) to be the product of the primes which divide \(n\). Then the \(abc\) conjecture of Masser-Oesterlé asserts that for all \(\varepsilon>0\) there is a real \(c_ 1>0\) such that for all relatively prime \(a,b,c\in\mathbb{Z}\) with \(a+b=c\), \(\max(| a|,| b|,| c|)\leq c_ 1 G(abc)^{1+\varepsilon}\). This is a quite strong conjecture which implies, among other things, the asymptotic Fermat conjecture.
This paper proves that if \(G=G(abc)\) and \(| c|>2\), then there exists an absolute constant \(c_ 2\) such that \[ \log\max(| a|,| b|,| c|)<G^{2/3+c_ 2/\log\log G}. \] This is a strengthening of the inequality \(\log\max(| a|,| b|,| c|)<c_ 3 G^{15}\) proved by C. L. Stewart and R. Tijdeman [Monatsh. Math. 102, 251-257 (1986; Zbl 0597.10042)]. However, it is still not strong enough for applications.
The proof is based on Baker’s theory on linear forms in logarithms; therefore the constant \(c_ 2\) can be explicitly computed.
Reviewer: P.Vojta (Berkeley)

MSC:
11J25 Diophantine inequalities
11J86 Linear forms in logarithms; Baker’s method
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References:
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