The well known abc conjecture asserts that for any coprime triple of positive integers satisfying a+b=c, we have c<K_ε rad(abc)^{1+ε}, where rad is the squarefree radical function.

In this talk, I will discuss a proof giving the first power-saving improvement over the trivial bound for the number of exceptions to this conjecture. The proof is based on a combination of various methods for counting rational points on curves, and a combinatorial analysis to patch these cases together. This is joint work with Tim Browning and Jared Lichtman.

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