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Not As Easy As ABC

April 5, 2020

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.

  1. Is the proof understood by the experts?

  2. Has the proof been generalized?

  3. Have new proofs been found?

  4. 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?

[Fixed typo]

13 Comments leave one →
  1. April 5, 2020 10:04 pm

    Apply the glorious PCP theorem to find?

  2. April 6, 2020 12:51 am

    There is some typo in ‘Have new the proofs been found?’, I don’t understand what you mean.

    • rjlipton permalink*
      April 6, 2020 12:19 pm

      Dear domotorp:

      I will fix. Please delete “the”.


      Stay well


  3. E.L. Wisty permalink
    April 6, 2020 6:07 am

    Reblogged this on Pink Iguana and commented:
    Stunningly lucid restatement of the ABC conjecture

    • rjlipton permalink*
      April 6, 2020 12:18 pm

      Dear E.L. Wisty:

      Thank you so much. Glad you liked the ABC statement. Wow “lucid” is a very nice term.


      Stay well


  4. anonymous permalink
    April 6, 2020 11:57 am

    Have you read Peter Scholze’s remark about the claimed proof of the ABC conjecture here? I do not understand why you say that at Woit’s blog only “personal factors” are discussed. Scholze (Field medalist) talks about mathematics.

  5. Madeleine Birchfield permalink
    April 7, 2020 12:18 am

    If Mochizuki had been releasing his work piecemeal in the last two decades since he started working on it circa 2002 instead of publishing it in one big chunk in 2015 it would have given the mathematics community enough time to digest his Interuniversal Teichmueller theory and give Mochizuki feedback and correct any potential logical or mathematical errors and clear up any confusions. The large roadblock that is Corollary 3.12 would certainly have been discovered earlier and the mathematics community prepared for Mochizuki’s arguments. Unfortunately, due to how it was all handled, it would take a while until the wider number theory and arithmetic geometry community to translate the results of Interuniversal Teichmueller Theory into traditional number theoric terminology and judge the logical correctness of Mochizuki’s work. Thankfully there have been recent progress in that direction by Dimitrov and Dupuy, but the controversey and damage is already done.

    Mochizuki said that IUT was only going to be useful for proving the abc conjecture and not much else. Personally I believe that Mochizuki is wrong about both IUT and the abc conjecture; that IUT will become an integral part of future research in arithmetic and anabelian geometry but will not be powerful enough to solve the abc conjecture (that Corollary 3.12 is wrong).

    • April 15, 2020 10:40 am

      interesting but actually am somewhat skeptical of one claim here about social dynamics. suppose mochizuki released intermediate papers. agree that is a good idea, and preferred, and the conventional/ accepted approach. however, there is an assumption that there would be an audience, that the (sometimes) imaginary “community” would necessarily respond. the audience for this research is very narrow.

      it appears mochizukis problem is more what RJL is pointing out in this essay. (4) does the proof have a clear roadmap? in this post RJL quite diplomatically tiptoes around the “elephant in the room” and stops short of explicitly accusing mochizuki of outright failure in this area, but lets just read between the lines and fill in the blanks. judging by many responses, mochizuki has a style that is hard to follow even by what might be called “anointed insiders”. my feeling is that the same concepts might be presentable in a more straightfwd way. unfortunately this is not exactly the same as proof correctness, they are orthogonal aspects of proof. there is a concept of proof *style*. his style is opaque, not transparent, that is the consensus, not always presented exactly that way, but supported by the many complaints, a strong underlying theme.

      is scholze also pointing to corollary 3.12? he has a extended comment on woits blog mentioning corollary 21.2

      idea on reading this: the original ABC conjecture is a “boiled down” idea. from wikipedia: ” The abc conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the Szpiro conjecture about elliptic curves.[1]” in other words, abstract, complex, high level theory boils down into a simple statement. now my idea, (1) is it possible to “boil down” some implications of the Mochizuki corollary in question into a straightfwd statement that does not involve extremely high level concepts. (2) then could this “boiled down” version be studied by some atypical means such as computational experiments to find “statistical evidence” one way or the other via large search spaces.

  6. Guy Haworth permalink
    April 8, 2020 5:59 am

    OK; the phrase ‘as easy as ABC’ is out the window

  7. April 14, 2020 9:04 pm

    woit has some nice analysis/ summary. he criticizes fesenko for criticizing the proofs naysayers. you conveniently omit this raging controvesy.

    fesenko is convinced the proof is correct. appreciate his essay but think he should work more on communicating it to other mathematicians instead of talking so much about revolutions. or at least, try to do both at the same time?

    ok, think fesenko is a great popsci writer and esp found this sentence notable

    While the range of activities where humans are better than
    AI continues to narrow, pure mathematics is most likely to be one of the last to be completely passed over to AI.

    there are some fairly rare experiments involving computational number theory, but suspect a few are attempting to apply AI even to number theory and other areas of pure mathematics. he is right, it seems to be one of the very last areas of encroachment by AI (somewhat similar to programming), but dont think this will hold a lot longer. at least measured in decades. which in a way is the “short run” for scientific progress…. am hoping to see new breakthroughs on the horizon where even mathematical proofs yield to AI, have worked in the area quite a few yrs now, have a little )( to show for it…


  1. Rjlipton’s thoughts on the possible ABC Conjecture proof – nebusresearch
  2. Se anuncia la publicación de la supuesta demostración de Mochizuki de la conjetura ABC (sí, tiene truco) - Gaussianos

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