Ellenberg, Jordan S. Congruence \(ABC\) implies \(ABC\). (English) Zbl 0986.11019 Indag. Math., New Ser. 11, No. 2, 197-200 (2000). 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)]. Reviewer: Roelof J.Stroeker (Rotterdam) Cited in 1 ReviewCited in 2 Documents MSC: 11D04 Linear Diophantine equations Keywords:\(abc\)-conjecture; congruence \(abc\)-conjecture Citations:Zbl 0668.10024 PDFBibTeX XMLCite \textit{J. S. Ellenberg}, Indag. Math., New Ser. 11, No. 2, 197--200 (2000; Zbl 0986.11019) Full Text: DOI arXiv 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.