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Congruence \(ABC\) implies \(ABC\). (English) Zbl 0986.11019

The \(abc\)-conjecture of Masser and Oesterlé states that for any \(\varepsilon> 0\) there exists a positive constant \(c_\varepsilon\) such that if \(a,b,c\) are coprime integers with \(a+b+c=0\) then \[ \max \{|a|,|b|,|c|\}< c_\varepsilon (\text{rad}(abc))^{1+\varepsilon}. \] Here the radical \(\text{rad}(n)\) is the largest squarefree divisor of \(n\). The congruence \(abc\)-conjecture for the integer \(N\) states that the \(abc\)-conjecture holds for all \(a,b,c\) (coprime and \(a+b+c=0\)) with \(abc\) divisible by \(N\).
In this paper the author shows that, for every integer \(N\), the congruence \(abc\)-conjecture for \(N\) implies the full \(abc\)-conjecture. This extends an observation of J. Oesterlé [Nouvelle approches du théorème de Fermat, Sémin. Bourbaki, Exp. No. 694, Astérisque 161/162, 165-186 (1988; Zbl 0668.10024)].

MSC:

11D04 Linear Diophantine equations

Citations:

Zbl 0668.10024
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References:

[1] Mazur, B. — Lectures on the ABC conjecture. In preparation.; Mazur, B. — Lectures on the ABC conjecture. In preparation.
[2] Oesterlé, J., Nouvelle approches du théorème de Fermat, Astérisque, 165-186 (1988) · Zbl 0668.12005
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