Not As Easy As ABC

Gödel’s Lost Letter and P=NP 2020-04-06

Is the claimed proof of the ABC conjecture correct?

[ Photo courtesy of Kyodo University ]

Shinichi Mochizuki is about to have his proof of the ABC conjecture published in a journal. The proof needs more than a ream of paper—that is, it is over 500 pages long.

Today I thought we would discuss his claimed proof of this famous conjecture.

The decision to published is also discussed in an article in Nature. Some of the discussion we have seen elsewhere has been about personal factors. We will just comment briefly on the problem, the proof, and how to tell if a proof has problems.

The Problem

Number theory is hard because addition and multiplication do not play well together. Adding numbers is not too complex by its self; multiplication by its self is also not too hard. For those into formal logic the theory of addition for example is decidable. So in principle there is no hard problem that only uses addition. None. A similar point follows for multiplication.

But together addition and multiplication is hard. Of course Kurt Gödel proved that the formal theory of arithmetic is hard. It is not complete, for example. There must be statements about addition and multiplication that are unprovable in Peano Arithmetic.

The ABC conjecture states a property that is between addition and multiplication. Suppose that

$\displaystyle A + B = C,$

for some integers ${1 \le A \le B \le C}$ . Then

$\displaystyle C \le ABC$

is trivial. The ABC conjecture says that one can do better and get

$\displaystyle C \le F(ABC),$

for a function ${F(X)}$ that is sometimes much smaller than ${X}$ . The function ${F(X)}$ depends not on the size of ${X}$ but on the multiplicative structure of ${X}$ . That is the function depends on the multiplicative structure of the integers. Note, the bound

$\displaystyle C \le ABC$

only needed that ${A,B,C}$ were numbers larger than ${1}$ . The stronger bound

$\displaystyle C \le F(ABC),$

relies essentially on the finer structure of the integers.

Roughly ${F(X)}$ operates as follows: Compute all the primes ${p}$ that divide ${X}$ . Let ${Q}$ be the product of all these primes. Then ${F(X) \le Q^{2}}$ works:

$\displaystyle C \le Q^{2}.$

The key point is: Even if ${p^{100}}$ , for example, divides ${X}$ , we only include ${p}$ in the product ${Q}$ . This is where the savings all comes from. This is why the ABC conjecture is hard: repeated factors are thrown away.

Well not exactly, there is a constant missing here, the bound is

$\displaystyle C \le \alpha Q^{2}$

where ${\alpha>0}$ is a universal constant. We can replace ${Q^{2}}$ by a smaller number—the precise statement can be found here. This is the ABC conjecture.

The point here is that in many cases ${F(ABC)}$ is vastly smaller than ${ABC}$ and so that inequality

$\displaystyle C \le F(ABC),$

is much better than the obvious one of

$\displaystyle C \le ABC.$

For example, suppose that one wishes to know if

$\displaystyle 5^{z} = 2^{x} + 3^{y},$

is possible. The ABC conjecture shows that this cannot happen for ${z}$ large enough. Note

$\displaystyle F(2^{x} 3^{y} 5^{z}) = 30$

for positive integers ${x,y,z}$ .

Is He Correct?

Eight years ago Mochizuki announced his proof. Now it is about to be published in a journal. He is famous for work in part of number theory. He solved a major open problem there years ago. This gave him instant credibility and so his claim of solving the ABC conjecture was taken seriously.

For example, one of his papers is The Absolute Anabelian Geometry of Canonical Curves. The paper says:

How much information about the isomorphism class of the variety ${X}$ is contained in the knowledge of the étale fundamental group?

A glance at this paper shows that it is for specialists only. But it does seem to be math of the type that we see all the time. And indeed the proof in his paper is long believed to be correct. This is in sharp contrast to his proof of the ABC conjecture.

Indicators of Correctness

The question is: Are there ways to detect if a proof is (in)correct? Especially long proofs? Are there ways that rise above just checking the proof line by line? By the way:

The length of unusually long proofs has increased with time. As a rough rule of thumb, 100 pages in 1900, or 200 pages in 1950, or 500 pages in 2000 is unusually long for a proof.

There are some ways to gain confidence. Here are some in my opinion that are useful.

Is the proof understood by the experts?
Has the proof been generalized?
Have new the proofs been found?
Does the proof have a clear roadmap?

The answer to the first question (1) seems to be no for the ABC proof. At least two world experts have raised concerns—see this article in Quanta—that appear serious. The proof has not yet been generalized. This is an important milestone for any proof. Andrew Wiles famous proof that the Fermat equation

$\displaystyle x^{p} + y^{p} = z^{p},$

has no solutions in integers for ${xyz \neq 0}$ and ${p \ge 3}$ a prime has been extended. This certainly adds confidence to our belief that it is correct.

Important problems eventually get other proofs. This can take some time. But there is almost always success in finding new and different proofs. Probably it is way too early for the ABC proof, but we can hope. Finally the roadmap issue: This means does the argument used have a nice logical flow. Proofs, even long proofs, often have a logic flow that is not too complex. A proof that says: Suppose there is a object ${X}$ with this property. Then it follows that there must be an object ${Y}$ so that ${\dots}$ Is more believable than one with a much more convoluted logical flow.

Open Problems

Ivan Fesenko of Nottingham has written an essay about the proof and the decision to publish. Among factors he notes is “the potential lack of mathematical infrastructure and language to communicate novel concepts and methods”—noting the steep learning curve of trying to grasp the language and framework in which Mochizuki has set his proof. Will the decision to publish change the dynamics of this effort?