Sorry for the silence of this blog for the last few months. This spring, my mother died, and I was very depressed about it. Depression is a difficult thing, and it left me without the energy or drive to do the difficult work of writing this kind of material. I’m trying to get back into the cycle of writing. I’m trying to make some progress in writing about type theory, but I’m starting with a couple of easier posts.

In the time when I was silent, I had a couple of people write to me to ask me to explain something called *the ABC conjecture*.

The ABC conjecture is a mathematical question about number theory that was proposed in the 1980s – so it’s relatively new as number theory problems go. It’s gotten a lot of attention recently, due to an almost soap-operatic series of events.

It’s a very hard problem, and no one had made any significant progress on it until about five years ago, when a well respected Japanese mathematician named Shinichi Mochizucki published a series of papers containing a proof of the conjecture.

Normally, when a proof of a hard problem gets published, mathematicians go nuts! Everyone starts poring over it, trying to figure it out, and see if it’s valid. That’s what happened the previous time someone thought they’d prooved it. But this time, no one has been able to make sense out of the proof!

The problem is that in order to build his proof, professor Mochizucki created a whole new mathematical theory, called inter-universal Teichmüller theory. The entire ABC conjecture proof is built in this new theory, and no one other than professor Mochizucki himself understands Teichmüller theory. Before anyone else can actually follow the proof, they need to understand the theory. Professor Mochizucki is a bit of a recluse – he has declined to travel anywhere to teach his new mathematical system. So in the five years since he first published it, no one has been able to understand it well enough to determine whether or not the proof is correct. One error in it was found, but corrected, and the whole proof remains in question.

Exactly why the proof remains unchecked after five years is a point of contention. Lots of mathematicians are angry at Professor Mochizucki for not being willing to explain or teach his theory. A common statement among critics is that if you create a new mathematical theory, you need to be willing to actually explain it to people: work with a group of mathematicians to teach it to them, so that they’ll be able to use it to verify the proof. But Professor Mochizuchki’s response has been that he *has* explained it: he’s published a series of papers describing the theory. He doesn’t want to travel and take time away from his work for people who haven’t been willing to take the time to read what’s he’s written. He’s angry that after five years, no one has bothered to actually figure out his proof.

I’m obviously not going to attempt to weigh in on whether or not Professor Mochizuki’s proof is correct or not. That’s so far beyond the ability of my puny little brain that I’d need to be a hundred times smarter before it would even be laughable! Nor am I going to take sides about whether or not the Professor should be travelling to teach other mathematicians his theory. But what I can do is explain a little bit about what the ABC conjecture is, and why people care so much about it.

It’s a conjecture in number theory. Number theorists tend to be obsessed with prime numbers, because the structure of the prime numbers is a huge and fundamental part of the structure and behavior of numbers as a whole. The ABC conjecture tries to describe one property of the structure of the set of prime numbers within the system of the natural numbers. Mathematicians would love to have a proof for it, because of what it would tell them about the prime numbers.

Before I can explain the problem, there’s a bit of background that we need to go through.

- Any non-prime number N is the product of some set of prime numbers. Those numbers are called the
*prime factors*of N. For example, 8 is 2×2×2 – so the set of prime factors of 8 is {2}. 28 is 2×2×7, so the prime factors of 28 are {2, 7}. 360 = 8 × 45 = 2×2×2×(9×5) = 2×2×2×3×3×5, so the prime factors of 360 are {2, 3, 5}. - For any number N, the
*radical*of N is product of its set of prime factors. So the radical of 8 (written rad(8)) is 2; rad(14)=14; rad(28)=14; rad(36)=6, rad(360)=30, etc. - Given two positive integers N and M, N and M are coprime if they have no common prime factors. A tiny bit more formally, if pf(N) is the set of prime factors of N, and M and N are coprime if and only if pf(N) ∩ pu(M) = ∅. (Also, if M and N are coprime, then rad(M×N) = ram(M)×rad(N).)

The simplest way of saying the ABC conjecture is that for the *vast majority* of integers A, B, and C, where A + B = C and A and B are coprime, C must be smaller than rad(A*B).

Of course, that’s hopelessly imprecise for mathematicians! What does “the vast majority” mean?

The usual method at times like these is to find some way of characterizing the size of the relative sizes of the set where the statement is true and where the statement is false. For most mathematicians, the sizes of sets that are interesting are basically 0, 1, finite, countably infinite, and uncountably infinite. For the statement of the ABC conjecture, they claim that the set of values for which the statement is true is infinite, but that the set of values for which it is false are finite. Specifically, they want to be able to show that the set of numbers for which rad(A*B)>C is finite.

To do that, they pull out a standard trick. Sadly, I don’t recall the proper formal term, but I’ll call it *epsilon bounding*. The idea is that you’ve got a statement S about a number (or region of numbers) N. You can’t prove your statement about N specifically – so you prove it about regions around N.

As usual, it’s clearest with an example. We want to say that C > rad(A*B) for most values of A and B. The way we can show that is by saying that for any value ε, the set of values (A, B, C) where A and B are coprime, and A + B = C, and rad(A*B) > C + ε is finite.

What this formulation does is give us a formal idea of how rare this is. It’s *possible* that there are some values for A and B where rad(A*B) is bigger that 1,000,000,000,000,000,000 + C. But the number of places where that’s true is finite. Since the full system of numbers is infinite, that means that in the overwhelming majority of cases, rad(A*B) < C. The size of the set of numbers where that's not true is so small that it might at well be 0 in comparison to the size of the set of numbers where it is true.
Ultimately, it seems almost trivial once you understand what the conjecture is. It's nothing more that the hypothesis that that if A + B = C, then most of the time, pf(A)*pf(B) < C. Once you've got that down, the question is, what's the big deal? Professor Mochuzuki developed five hundred pages of theory for *this*? People have spent more than five years trying to work through his proof just to see if it’s correct for a statement like this? Why does anybody care so much?

One answer is: mathematicians are crazy people!

The better answer is that simple statements like this end up telling us very profound things about the deep structure of numbers. The statements reduce to something remarkably simple, but the meaning underneath it is far more complex than it appears.

Just to give you one example of what this means: If the conjecture is true, then there’s a three-line proof of Fermat’s last theorem. (The current proof of Fermat’s last theorem, by Andrew Wiles, is over 150 pages of dense mathematics.) There’s quite a number of things that number theoreticians care about that would fall out of a successful proof.

Welcome back. I’ve missed you.

Seconded. Welcome back, and thanks for taking the time to write a little about mathematics!

Nitpick: plenty of people understand Teichmüller theory; it’s Mochizuki’s version that parameterizes number field structures on elliptic curves (AIUI) that nobody but him understands well enough yet.

Welcome back. I’m very sorry for your loss.

The new post is wonderful.

In “background, #3”

rad(M×N) = ram(M)×rad(N)

Should probably be:

rad(M×N) = rad(M)×rad(N).

I don’t think your assessment that nobody understands IUTeich, and that Mochizuki is unwilling to help teach it, is entirely valid. The report published maybe a month ago here: http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTeich%20Verification%20Report%202014-12.pdf shows that he does indeed work to help (quite a small audience that is willing to spend their time understanding the conjecture) promote understanding, and that some well-regarded mathematicians are working to understand his work.

Hi, Mark. I’m so sorry about your mom.

On the “vast majority” thing: the term I’d always heard used for this was “almost all”, which meant “all but a finite number in an infinite domain.” I always liked the term because it was well defined, but sounded like it wasn’t.

I smiled when my RSS feed for this blog was in bold for the first time in ages. Really good to see you posting again.

Very sorry to hear about your mum.

Um, ok are there any examples of A, B and C such that rad(A*B)>C? Is there any way to find them other than brute force?

First of all, you’re looking for rad(A*B) B = 1

E = 2 => B = 3

E = 3 => B = 7

E = 4 => B = 3*5

E = 5 => B = 31

E = 6 => B = 3^2*7

E = 7 => B = 127

E = 8 => B = 3*5*17

E = 9 => B = 7*73

E = 10 => B = 3*11*31

As we see, we only get a repeated prime for E = 6 so far. For the rest we’re going to get a much larger value of rad(A*B).

Ugh; looks like the majority of that comment got mangled 🙁

Really glad to see you’re feeling up to posting again! And sorry you had to go through so much to get to the other side.

I am curious about the 3 lines proof of Fermat’s last theoreme, can you post it? Could it be the same proof that Fermat’s could not write on his notes?

The assertion that ABC yields a 3 line proof of FLT is false.

What is true is that ABC yields a 3-line proof of FLT for all sufficiently

large exponents. However, the proof does not yield an actual value

for what “sufficiently large” means.