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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 $abc$-conjeture 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 {\it 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].

11D99Diophantine equations