The abc Conjecture

The mathematical internets are abuzz with the news that the Japanese mathematician Shinichi Mochizuki has released a proof of the famous abc Conjecture.

What’s the conjecture?   Well, as we learned in preschool, we know every positive integer has a unique factorization into primes.  If n is a positive integer, let’s write rad(n) for the product of the primes which appear in the factorization, but if a prime appears more than once we use it only once.  For example rad(12)=rad(2*2*3)=6 (here * is multiplication).

Say a,b, and c are three positive numbers.  We will say that these three numbers are “neighbors” if c < rad(a*b*c).

Say a,b, and c are three positive numbers which have no primes in common in their prime factorizations.  And also say you’ve picked them so that a+b=c. Evidence suggests that almost all of these triples of numbers  are neighbors.  The abc Conjecture says that in fact there are very few such triples which fail to be neighbors:

Let \epsilon>0 be any fixed number (pick one as small as you like!). Then there only are finitely many triples of positive integers, a, b, and c, such that a,b,c have no common prime factors and

c > rad(a*b*c)^{1+\varepsilon}.

Like most number theory problems (see Fermat’s Last Theorem!), it is much easier to explain the problem then it is to find the solution.  Mochizuki has released four papers — all long and full of high powered math — which prove that the abc Conjecture is true, and does much, much more.

The reason people are excited is that the abc Conjecture is closely related to a number of other very interesting questions and conjectures in number theory.  Check out the wikipedia page for a sampling.  And, of course, proving a deep new result in math usually involves developing deep new mathematics.  And that opens new doors and raises new questions!

It’s all very exciting, but be warned!  Right now it seems that only Mochizuki understands his proof, and it’s going to take even the world experts a long time to read everything and look for mistakes.  As grp wrote on mathoverflow:

…there are plenty of top experts in arithmetic geometry who are presently struggling to get even a small handle on what is really going on in Mochizuki’s papers.

— grp on mathoverflow

Part of the reason is that Mochizuki breaks really new mathematical ground* in his papers and it will take even the experts quite awhile to get a grasp on what all he’s done.

If you’d like a taste of the math in Mochizuki’s work and a sense of the interest among research mathematicians in his work, check out the discussion on mathoverflowAlso take a look at Jordan Ellenberg’s blog post.

We’ll let you know when we hear if the experts give Mochizuki’s proof the thumbs up/down.

*  Wiles’ proof of Fermat’s Last Theorem and Perelman’s proof of Thurston’s Geometrization Conjecture developed a lot of deep new mathematics, but it was in the context of well known techniques.  Experts in those areas could much more quickly get up to speed based on their prior knowledge of these techniques.