The \(abc\)-conjecture (Q1314951)

The Masser-Oesterlé \(abc\) conjecture asserts that, given \(\varepsilon>0\), for all triples \(a\), \(b\), \(c\) of integers with \(a+b+c=0\) and \((a,b,c)=1\), we have \(\max( | a|,| b|,| c|)\ll_ \varepsilon \prod_{p\mid abc} p^{1+\varepsilon}\). First, Lang motivates this conjecture by stating and proving the analogue, Mason's theorem, in the function field case. He then shows how Mason's theorem and the \(abc\) conjecture (would) imply, respectively, an analogue of the Fermat conjecture over function fields, and the Fermat conjecture over number fields for sufficiently large exponent (``asymptotic Fermat''). The paper also discusses how the \(abc\) conjecture relates to a conjecture of M. Hall on \(| y^ 2- x^ 3|\) and to the generalized Szpiro conjecture.