The abc conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory, first proposed by Joseph Oesterlé (1985) and David Masser (1988). It is stated in terms of three positive integers, a, b and c (hence the name) that are relatively prime and satisfy a + b = c. If d denotes the product of the distinct prime factors of abc, the conjecture essentially states that d is usually not much smaller than c. In other words: if a and b are composed from large powers of primes, then c is usually not divisible by large powers of primes. The precise statement is given below.
The abc conjecture has already become well known for the number of interesting consequences it entails. Many famous conjectures and theorems in number theory would follow immediately from the abc conjecture. Goldfeld (1996) described the abc conjecture as “the most important unsolved problem in Diophantine analysis“.
Lucien Szpiro attempted a solution in 2007, but it was found to be incorrect. In August 2012 Shinichi Mochizuki posted his four preprints which develop a new inter-universal Teichmüller theory, with an application to the proof of several famous conjectures including the abc conjecture. His papers were submitted to a mathematical journal and are being refereed, while various activities to study his theory have been run.
In August 2012, Shinichi Mochizuki released a series of four preprints on Inter-universal Teichmuller Theory which is then applied to prove several famous conjectures in number theory, including the Szpiro’s conjecture, the hyperbolic Vojta’s conjecture and the abc conjecture. Mochizuki calls the theory on which this proof is based “inter-universal Teichmüller theory (IUT)”. The theory is radically different from any standard theories, it goes outside the scope of arithmetic geometry. The theory was developed over two decades, the last four IUT papers occupy the space of 500 pages and use many of his prior published papers.
An error in the last of the articles was pointed out by Vesselin Dimitrov and Akshay Venkatesh in October 2012, and Mochizuki revised appropriate parts of his papers on “inter-universal Teichmüller theory“. Mochizuki has refused all requests for media interviews, but released progress reports in December 2013 and December 2014. He has invested hundreds of hours to run seminar and meetings to discuss his theory. According to Mochizuki, verification of the core proof is “for all practical purposes, complete.” However, he also stated that an official declaration shouldn’t happen until some time later in the 2010s, due to the importance of the results and new techniques. In addition, he predicts that there are no proofs of the abc conjecture that use significantly different techniques than those used in his papers. The first international workshop on Mochizuki’s theory was organized by Ivan Fesenko and held in Oxford at in December 2015. It helped to increase the number of mathematicians who had thoroughly studied parts of the IUT papers or related prerequisite papers to a two digital one. The next workshop on IUT Summit will be held at the Research Institute for Mathematical Sciences in Kyoto in July 2016. There are several introductory texts and surveys of the theory, written by Mochizuki and other mathematicians.