Proven! — 'most important unsolved problem' in numbers

[drm604]

2. This is big if the proof is correct.

http://www.maa.org/mathland/mathtrek_12_8.html

Astonishingly, a proof of the ABC conjecture would provide a way of establishing Fermat's last theorem in less than a page of mathematical reasoning. Indeed, many famous conjectures and theorems in number theory would follow immediately from the ABC conjecture, sometimes in just a few lines.

"The ABC conjecture is amazingly simple compared to the deep questions in number theory," says Andrew J. Granville of the University of Georgia in Athens. "This strange conjecture turns out to be equivalent to all the main problems. It's at the center of everything that's been going on."

"Nowadays, if you're working on a problem in number theory, you often think about whether the problem follows from the ABC conjecture," he adds.

"The ABC conjecture is the most important unsolved problem in Diophantine analysis," Goldfeld writes in Math Horizons. "It is more than utilitarian; to mathematicians it is also a thing of beauty. Seeing so many Diophantine problems unexpectedly encapsulated into a single equation drives home the feeling that all the subdisciplines of mathematics are aspects of a single underlying unity, and that at its heart lie pure language and simple expressibility."