Mathematician Solved a Nearly Impossible Maths Problem

December 30, 2015 | Joanne Kennell

Math equation
Photo credit: Tom Brown/Flickr (CC by SA 2.0)

But can it be considered solved if only he understands it?

So you claim to know how to solve a famous conjecture?  Prove it!  That is exactly what Japanese mathematician, Shinichi Mochizuki of the Research Institute for Mathematical Sciences from Kyoto University, has done.  However, no one else understands it.

Back in August 2012, Mochizuki posted a series of four papers on his personal website claiming to prove the ABC conjecture — an important problem in number theory.  The ABC conjecture states that given three positive integers, a, b, c — where a + b = c, and where each integer has no prime factors in common — and given d, which denotes the product of the distinct prime factors, then there are only a finite set of triple integers where d is actually smaller than c.

Confused yet?  Here is an example.  Let a = 16, b = 21, and c = 37.  The prime factor of 16 is 2, prime factors of 21 are 3 and 7, and the prime factor of 37 is 37 — all unique.  Therefore, d = 2 * 3 * 7 * 37 = 1554, which is larger than c.  According to mathematical experts, this happens all the time and there is a lot of numerical evidence to support the ABC conjecture, however it has never been proven mathematically.  Until now.

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Mochizuki uses a process that he calls Inter-Universal Teichmüller Theory — an arithmetic version of Teichmüller theory for number fields.  He also includes new terminology and definitions invented by him (there is something called a Frobenioid, for example) to explain his proof.  Unfortunately, most people who have attempted to read and understand his theory have given up. However, some experts are still giving this solution the benefit of the doubt since Mochizuki has an esteemed reputation in the mathematics community.

Unfortunately, Mochizuki refuses to speak to the press or travel to discuss his work.  The Clay Mathematics Institute and the Mathematical Institute at Oxford recently sponsored a meeting about his work including many world experts in number theory.  Mochizuki did not attend.  The goal of the meeting was to give these experts some of the necessary background and information to begin to read through the four papers.  However, the meeting ended in frustration, since the audience repeatedly asked for illustrative examples but none were ever given.

What does this mean for Mochizuki’s proof?  Although there are a very small number of mathematicians that claim to have read and verified the theory, a much larger group remains completely perplexed.  Until there is someone who can translate the results in a way that the majority of mathematicians can understand, the status of the ABC conjecture will remain debatable.  There has to be a version that is more widely understood before the conjecture can be considered solved.

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