Congruence \(ABC\) implies \(ABC\) (Q5935897)

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 NEWLINE\[NEWLINE\max \{|a|,|b|,|c|\}< c_\varepsilon (\text{rad}(abc))^{1+\varepsilon}.NEWLINE\]NEWLINE 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 \(N\). NEWLINENEWLINENEWLINEIn 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 \textit{J. Oesterlé} [Nouvelle approches du théorème de Fermat, Sémin. Bourbaki, Exp. No. 694, Astérisque 161/162, 165-186 (1988; Zbl 0668.10024)].