Some blogs/articles claim that many problems will follow immediately from the ABC theorem if it is indeed a theorem. For example Fermat's last theorem will follow in a one-page proof from the abc theorem.

My question is: can someone use the abc theorem to outline some proofs of open problems, or solved problems with difficult proofs, that are solved immediately using the abc theorem? It would be best if there is one solved proof per answer.

I am also especially interested in seeing a one-page proof of Fermat's last theorem using the ABC theorem!