An exciting story has developed over the past few months. In August 2012, after months of rumors, the mathematician Shinichi Mochizuki rounded out a series of papers which he claims prove the ABC Conjecture:
For every ε > 0, there are only finitely many triples of coprime positive integers a + b = c such that c > d^(1+ε), where d denotes the product of the distinct prime factors of abc.
This easy to state conjecture has many important consequences in number theory. Professor Mochizuki has garnered an abundance of coverage from the popular media in the ensuing months due to the novelty of his approach; it is unclear (assuming correctness) how long it would take for his proof to be accepted by the mathematical community.
If you would like to learn more about this story, I have selected a few articles:
- MathOverflow – Philosophy behind Mochizuki’s work on the ABC conjecture
- Nature – Proof claimed for deep connection between primes
- Boston Globe – An ABC proof too tough even for mathematicians
For complete coverage and technical details, check this article on the PolyMath wiki.
You may also find interesting this recent article from Kevin Hartnett, author of the Boston Globe story linked above. It’s the slightly depressing account of a man who, like Professor Mochizuki, disappeared from the mathematical community to tackle a big problem–Goldbach’s conjecture.
