The \(abc\) conjecture
The \(ABC\)-conjecture suggests that if \(A+ B= C\) in positive integers, then \(C\) cannot be small compared to the `radical', or `conductor', \(N= N(ABC)\); here \(N(D):= \prod_{p\mid D} p\). More precisely, the allegation is that for every \(\varepsilon> 0\) there is uniformly a \(c= c(\varepsilon)\), depending only on \(\varepsilon\), so that \(C\leq c(\varepsilon) N^{1+ \varepsilon}\). It is then easy to show that \(\lim_{\varepsilon\to 0} c(\varepsilon)= \infty\). Whatever, it is interesting to consider variously the sup (or lim sup) of the ratios \(\alpha (A, B, C)= \log C/\log N\), or \(\rho (A, B, C)= \log ABC/ \log N\) as one ranges over relatively prime triples \(A+B=C\). The author is the discoverer of a large number of the extreme known cases (the record holders are \(2+ 3^{10} \cdot 109= 23^5\) with \(\alpha= 1.62991 \dots\) and \(13 \cdot 19^6+ 2^{30} \cdot 5= 3^{13}\cdot 11^2\cdot 31\) with \(\rho= 4.41901 \dots\), respectively noticed by Reyssat in 1987, and by the author in 1992). The present useful survey describes the conjecture for natural numbers \(A\), \(B\), \(C\); it recounts a substantial number of its implications, namely to a variety of well known diophantine problems (of course including Fermat's last theorem for exponent \(n\geq 5> 4.41901 \dots)\); it discusses techniques for numerically testing the conjecture; and it mentions various obvious generalizations.
scientific article; zbMATH DE number 915927