A Japanese mathematician claims to have solved one of the most important problems in his field. The trouble is, hardly anyone can work out whether he's right.

The biggest mystery in mathematics

This article in Nature is just wonderful. Everyone will find it interesting, but those in the Algebraic Number Theory class this fall will be particularly interested in the topic – by the way, it’s not too late to join the class. After spending some time over the summer looking at Category Theory, I’m tempted to tackle Mochizuki’s proof as I’m intrigued at new methods in mathematical thinking (and explaining.)

The *abc* conjecture refers to numerical expressions of the type *a + b = c*. The statement, which comes in several slightly different versions, concerns the prime numbers that divide each of the quantities *a*, *b* and *c*. Every whole number, or integer, can be expressed in an essentially unique way as a product of prime numbers — those that cannot be further factored out into smaller whole numbers: for example, *15 = 3 × 5* or *84 = 2 × 2 × 3 × 7*. In principle, the prime factors of *a* and *b* have no connection to those of their sum, *c*. But the *abc* conjecture links them together. It presumes, roughly, that if a lot of small primes divide *a* and *b* then only a few, large ones divide *c*.

Thanks to Rama for bringing this to my attention!