The ABC Conjecture

1985

The abc conjecture was first proposed in

Masser, D. W. (1985), "Open problems", in Chen, W. W. L., Proceedings of the Symposium on Analytic Number Theory, London: Imperial College

and states that

For every \(\varepsilon > 0\), there exist only finitely many triples \((a,b,c)\) of positive coprime integers, with \(a + b = c\), such that \(c>{\rm rad}(abc)^{1+\varepsilon}\)

(where the radical of \(n,\ {\rm rad}(n)\) means the product of the distinct prime factors of \(n\)).

It is important because many other results follow from it including Fermat's Last Theorem.

August 30, 2012 (updated several times in 2013, 2014 and 2015 too)

A proof was proposed on August 30, 2012 by Shinichi Mochizuki in the papers

http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20I.pdf

http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20II.pdf

http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20III.pdf

http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf

Many newspaper articles followed making it famous but it has not yet been verified and will take a while to do so.

Examples of these are:

September 10, 2012

Nature article on Mochizuki's proposed proof of ABC conjecture

September 12, 2012

Science magazine article on Mochizuki's proposed proof of ABC conjecture

September 17, 2012

New York Times article on Mochizuki's proposed proof of ABC conjecture

September 19, 2012

The Telegraph article on Mochizuki's proposed proof of ABC conjecture

November 4, 2012

Boston Globe article on Mochizuki's proposed proof of ABC conjecture