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Zbl 0973.11047
Baker, A.
Logarithmic forms and the $abc$-conjecture. (English)
[A] Gy\H{o}ry, 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). ISBN 3-11-015364-5/hbk

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 $abc$. \par The present paper contains sharpening of this conjecture: \par 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. \par 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.
[Maurice Mignotte (Strasbourg)]

MSC 2000:: *11D99 Diophantine equations

Keywords: $abc$-conjecture; linear forms in logarithms

Cited in: Zbl pre05959384

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