The bizarre story of a maths proof that is only true in Japan

Summary

A baffling proof of the ABC conjecture set off a bitter divide among mathematiciansAndreasG/Shutterstock As a science journalist, there are some stories that you return to again and again. There is climate change, of course – the biggest story of our time. Then there are the big questions, such as “what is consciousness?” or “are we alone in the universe?”, and so on. But one of the stories I simply cannot let go of is a mathematical argument that has been raging for over a decade. I call it “the proof that is only true in Japan”. Let me start at the beginning. In 2012, mathematician Shinichi Mochizuki at Kyoto University, Japan, published an extraordinary set of papers, across 500 pages, detailing what he called inter-universal Teichmüller (IUT) theory. This was a framework for stretching and deforming our ordinary concepts of mathematical objects, like numbers and their relationships through multiplication and addition, transferring them to new mathematical “universes” where they can be poked and prodded to yield new insights. “He is literally taking apart conventional objects in terrible ways and reconstructing them in new universes,” one mathematician told me at the time. That’s pretty cool-sounding, but what made this a particularly big deal was that Mochizuki was claiming to have used IUT to solve the abc conjecture, a deep problem in number theory that is now 40 years old. Unlike IUT, the statement of the conjecture is relatively easy to understand, as it simply starts with the equation a + b = c, where each letter stands for a whole number, or integer. You also need to know that every integer can be divided into its prime factors, the prime numbers that serve as the building blocks for all numbers. For example, the prime factors of 21 are 3 and 7, while for 12 they are 2, 2 and 3. For 12, 2 appears twice in the list of prime factors, so we can also talk about distinct prime factors, meaning we only list each number once. What the abc conjecture proposes is t...