As easy as ABC?

Here’s my latest news story for Nature. There’s a lot of superscripts in here, for which I’ll use the x**n notation. Every time I encounter mathematicians, I’m reminded what a very different world they live in.
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If it’s true, a Japanese mathematician’s solution to a conjecture about whole numbers would be an ‘astounding achievement’

The recondite world of mathematics is abuzz with a claim that one of the most important problems in number theory has been solved.

Japanese mathematician Shinichi Mochizuki of Kyoto University has released a 500-page proof of the ABC conjecture, which describes a purported relationship between whole numbers – a so-called Diophantine problem.

The ABC conjecture might not be as familiar to the wider world as Fermat’s Last Theorem, but in some ways it is more significant. “The ABC conjecture, if proved true, at one stroke solves many famous Diophantine problems, including Fermat's Last Theorem”, says Dorian Goldfeld, a mathematician at Columbia University in New York.

“If Mochizuki’s proof is correct, it will be one of the most astounding achievements of mathematics of the 21st century”, he adds.

Like Fermat’s theorem, the ABC conjecture is a postulate about equations of the deceptively simple form A+B=C that relate three whole numbers A, B and C. It involves the concept of a square-free number: one that can’t be divided by the square of any number. 15 and 17 are square free-numbers, but 16 and 18 – divisible by 4**2 and 3**2 respectively – are not.

The “square-free” part of a number n, denoted sqp(n), is the largest square-free number that can be formed by multiplying prime factors of n. For instance, sqp(18) = 2×3 = 6.

If you’ve got that, you should get the ABC conjecture. Proposed independently by David Masser and Joseph Oesterle in 1985, it concerns a property of the product of the three integers A×B×C, or ABC – or more specifically, of the square-free part of this product, which involves their distinct prime factors.

The conjecture states that the ratio of sqp(ABC)**r/C always has some minimum value greater than zero for any value of r greater than 1. For example, if A=3 and B=125, so that C=128, sqp(ABC)=30 and sqp(ABC)**2/C = 900/128. In this case, where r=2, sqp(ABC)**r/C is nearly always greater than 1, and always greater than zero.

It turns out that this conjecture encapsulates many other Diophantine problems, including Fermat’s Last Theorem (which states that A**n + B**n = C**n has no integer solutions if n>2). Like many Diophantine problems, it is at root all about the relationships between prime numbers – according to Brian Conrad of Stanford University, “it encodes a deep connection between the prime factors of A, B and A+B”.

“The ABC conjecture is the most important unsolved problem in Diophantine analysis”, says Goldfeld. “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.”

Unsurprisingly, then, many mathematicians have expended a great deal of effort trying to prove the conjecture. In 2007 the French mathematician Lucien Szpiro, whose work in 1978 led to the ABC conjecture in the first place, claimed to have a proof of it, but it was soon found to be flawed.

Like Szpiro, and also like Andrew Wiles who proved Fermat’s Last Theorem in 1994, Mochizuki has attacked the problem using the theory of elliptic curves, which are the smooth curves generated by algebraic relationships of the sort y**2 = x**3 + ax + b.

There, however, the relationship of Mochizuki’s work to previous efforts stops. In the present and earlier papers he has developed entirely new techniques that very few other mathematicians yet fully understand. “His work is extremely novel”, says Conrad. “It uses a huge number of new insights that are going to take a long time to be digested by the community.”

This novelty invokes entirely new mathematical ‘objects’ – abstract entities analogous to more familiar examples such as geometric objects, sets, permutations, topologies and matrices. “At this point he is probably the only one that knows it all”, says Goldfeld.

As a result, Goldfeld says, “if the proof is correct it will take a long time to check all the details.” The proof is spread over four long papers, each of which rests on earlier long papers. “It can require a huge investment of time to understand a long and sophisticated proof, so the willingness by others to do this rests not only on the importance of the announcement but also on the track record of the authors”, Conrad explains.

Mochizuki’s track record certainly makes the effort worthwhile. “He has proved extremely deep theorems in the past, and is very thorough in his writing, so that provides a lot of confidence”, says Conrad. And he adds that the payoff would be more than a matter of simply verifying the claim. “The exciting aspect is not just that the conjecture may have now been solved, but more importantly that the new techniques and insights he must have had to introduce should be very powerful tools for solving future problems in number theory.”

Mochizuki’s papers:
Paper 1
Paper 2
Paper 3
Paper 4