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Published on December 18th, 2012 | by Jo

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ABC theorem

For several years, a highly respected mathematician named Shinichi Mochizuki has not published much work. His colleagues thought he was probably working on something important and exciting, but they had no idea what it could be.

On 30 August, Shinichi released four documents, with a total of around 500 pages of mathematics. Shinichi’s writings include new tools and techniques, and different ways of looking at mathematical objects. Shinichi’s work was very long, and because so much of it is new, it is hard for other mathematicians to understand. However, there’s a good reason to get to grips with this new maths. Deep within the work is a powerful mathematical result – a proof of the ABC conjecture.

The ABC theorem describes the factors that numbers have. Factors are things that divide your number evenly – for example 4 is a factor of 32. Some factors also have factors, and you can break them down and down into their smallest parts – these are called prime factors. Some numbers have lots of prime factors, and some have very few – 32’s only prime factor is 2.

The ABC theorem says that if you have two large numbers with very few, very small prime factors, then their sum will have many large prime factors.

Before the papers are accepted as correct, several mathematicians will have to review them. This means those mathematicians have to understand every step in the papers, and make sure there are no mistakes. This can take some time, but in this case it could take years. Shinichi has spent years thinking about his new proofs, but he hasn’t been sharing his results regularly. Anyone wanting to understand Shinichi’s final result will have to first understand several years’ worth of preparation work.

It might be hard, but the checking is well underway. Several mathematicians are checking Shinichi’s maths. Two mathematicians, Vesselin Dimitrov and Akshay Venkatesh, have already found something not quite right in Shinichi’s documents. Shinichi has suggested there’s a way to fix the proof, and will distribute a new version soon.

Shinichi is trying his best to help other mathematicians understand his work. He gave a talk this month, and will publish a survey article, similar to a map or guide, of his ideas in March. Even with this extra assistance, it will still take a long time to know whether Shinichi’s proofs are correct.

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