Baker, A. Logarithmic forms and the \(abc\)-conjecture. (English) Zbl 0973.11047 Győry, Kálmán (ed.) et al., Number theory. Diophantine, computational and algebraic aspects. Proceedings of the international conference, Eger, Hungary, July 29-August 2, 1996. Berlin: de Gruyter. 37-44 (1998). The \(abc\)-conjecture asserts that if \(a\), \(b\), \(c\) are integers with \[ a+b+c=0, \quad \gcd(a,b,c)=1 \] then for any \(\varepsilon>0\), \[ \max\{|a|, |b|, |c|\} < C_\varepsilon N^{1+\varepsilon}, \] where \(N=\prod_{p, p |{abc}} p,\) is the conductor of \(abc\). The present paper contains sharpening of this conjecture: Conjecture 1. If \(a\), \(b\), \(c\) are integers satisfying (1) then, for any \(\varepsilon>0\), \[ \max\{|a|, |b|, |c|\} <C_1 (\varepsilon^{-\omega} N)^{1+\varepsilon}, \] where \(\omega\) denotes the number of distinct prime fators of \(abc\) and \(C_1\) is an absolute constant. Conjecture 2. There are absolute constants \(\kappa\) and \(C_2\) such that, if (1) holds then, for any \(\varepsilon>0\), \[ \max\{|a|, |b|, |c|\} <C_2 \varepsilon^{-\kappa\omega(ab)} N^{1+\varepsilon}. \] The author discusses in detail the links between these conjectures and natural conjectures on linear forms in logarithms, in the archimedean and nonarchimedean cases. He also presents several interesting observations about these conjectures.For the entire collection see [Zbl 0887.00013]. Reviewer: Maurice Mignotte (Strasbourg) Cited in 1 ReviewCited in 7 Documents MSC: 11D99 Diophantine equations Keywords:\(abc\)-conjecture; linear forms in logarithms PDFBibTeX XMLCite \textit{A. Baker}, in: Number theory. Diophantine, computational and algebraic aspects. Proceedings of the international conference, Eger, Hungary, July 29--August 2, 1996. Berlin: de Gruyter. 37--44 (1998; Zbl 0973.11047)