Shinichi Mochizuki of Kyoto University in Japan is claiming to have found proof (divided into four separate studies with 500+ pages) of the so-called abc conjecture, a longstanding problem in number theory which predicts that a relationship exists between prime numbers. Now the tricky part? Other mathematicians need to dig into his extensive work, and confirm that he’s right.
Philip Ball of Nature News explains the abc conjecture:
Like Fermat’s theorem, the abc conjecture refers to equations of the form a+b=c. It involves the concept of a square-free number: one that cannot be divided by the square of any number. Fifteen and 17 are square free-numbers, but 16 and 18 – being divisible by 4^2 and 3^2, respectively – are not.
The ‘square-free’ part of a number n, sqp(n), is the largest square-free number that can be formed by multiplying the factors of n that are prime numbers. For instance, sqp(18)=2×3=6.
If you’ve got that, then you should get the abc conjecture. It concerns a property of the product of the three integers axbxc, or abc – or more specifically, of the square-free part of this product, which involves their distinct prime factors. It states that for integers a+b=c, 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, then sqp(abc)=30 and sqp(abc)2/c = 900/128. In this case, in which r=2, sqp(abc)r/c is nearly always greater than 1, and always greater than zero.
If you don’t get any of that or what Mochizuki has done, don’t worry — many mathematicians don’t either. And in fact, Mochizuki is considered somewhat of a genius and a guy who’s in a league of his own. He thinks in terms of mathematical ‘objects’ — abstract entities like geometric objects, sets, permutations, topologies, and matricies. Ball quotes mathematician Dorian Goldfeld as saying, “At this point, he is probably the only one that knows it all.” Via Have mathematicians finally discovered the hidden relationship between prime numbers?.