abc Conjecture (Masser–Oesterlé)

Prove the abc conjecture of Masser and Oesterlé: For every ε > 0, there exists a constant Kε > 0 such that for all coprime integers a, b, and c with a + b = c, the radical of abc (the product of the distinct prime factors of abc) satisfies rad(abc) > Kε c^{1−ε}.

Background

The paper studies coprime integer solutions to a + b = c and the size of the radical rad(abc), defined as the product of the distinct prime divisors of abc. The abc conjecture asserts that rad(abc) cannot be much smaller than c for such solutions—more precisely, it should exceed a constant multiple of c^{1−ε} for any fixed ε > 0. This conjecture has far-reaching consequences in number theory, including implications for results such as Fermat’s Last Theorem, and remains unresolved in full generality.

The author presents elementary arguments showing that the abc inequality holds for almost all triples in a precise quantitative sense, bounding the number of exceptions within a cube by O(N^{2/3}). These results contextualize recent refinements but do not resolve the conjecture itself.