Folha: ‘Is the abc conjecture coming to an end?
Reproduction of Marcelo Viana’s column in Folha.
It’s the biggest mathematical controversy of our time and it’s been embittering the community for over a decade. But the way out may be coming. At the heart of it is a question about prime numbers.
Take two integers, a and b, greater than 1 and prime to each other, i.e. without common prime divisors, and consider c=a+b. It is obvious that the product of the three numbers, abc, is greater than any of them. In 1985, Joseph Oesterlé and David Masser conjectured something much more subtle: if we multiply only the distinct prime factors of abc the result will still be much larger than c, in a mathematically precise sense, except perhaps for a finite number of cases.
This “abc conjecture”, as it has come to be known, is a strange statement, even unexpected, because the prime factors of a sum c=a+b seem to have little to do with the prime factors of the respective parts. But it has attracted enormous interest in the mathematical community, largely because it has been shown that, if true, it leads to the solution of many important open problems in number theory. In particular, it will provide a new proof of Fermat’s theorem, probably simpler than the one given by Andrew Wiles in the 1990s.
Many mathematicians have worked on the question, but to this day there is no consensual answer accepted by the international community. The best-known and most controversial attempt is by Shinichi Mochizuki, from Kyoto University. In August 2012, he posted 4 papers on the internet, totaling over 500 pages, which would contain a proof of the conjecture. Unfortunately, experts have a hard time understanding Mochizuki’s arguments and he doesn’t have much patience for explaining them either. A few colleagues claim to have understood his arguments, but even these are unable to explain them satisfactorily to others.
I wrote about this here in 2018, when there was a coordinated effort to clarify the situation, including bringing some of the leading experts on the subject to Kyoto. But it was unsuccessful, which only increased skepticism. Eight years on, the impasse remains, but recently there has been a concrete expectation that technological advances can finally resolve the issue.
In 2023, a group of experts led by Kato Fumiharu began an international collaborative project to formalize Mochizuki’s arguments in Lean language: if they achieve a complete formalization, this will prove beyond a shadow of a doubt that the proof of the conjecture is correct. In a recent press conference, Fumiharu said that the group has already achieved a “deep understanding” of the ideas in the proof, but has not yet managed to overcome what is considered to be the most delicate point in all the reasoning.