Easy as ABC
A modern mathematical proof is not very different from a modern machine: the simple fundamental principles are hidden and almost invisible under a mass of technical details—Hermann Weyl.
Shinichi Mochizuki is a mathematician who is at the center of a decade old claim. He has—he says since 2012—solved a famous open problem in number theory called the abc conjecture. This conjecture is in number theory and would revolutionize our understanding of the structure of the natural numbers.
What is the ABC?
The abc conjecture is a conjecture in number theory, first proposed by Joseph Oesterle and David Masser independently around 1985. It is stated in terms of three positive integers, 
The common term for the product 
For every
, all but finitely many triples of relatively prime positive numbers giving
have
Intuitively what it says is that numbers that have large powers of different primes cannot be related by addition. As 
The point of the conjecture is that it relates addition and multiplication. It allows making inferences about the multiplicative structure of natural numbers from additive properties and vice-versa. The formal theory of the natural numbers with respect to addition alone, called Presburger arithmetic, is decidable, as is the theory of multiplication alone, called Skolem arithmetic. The theory of both 
ABC Implies What?
The conjecture has high explanatory power in that many other conjectures (listed here) follow from it. Among them, we note:
- Whereas no one has significantly simplified Andrew Wiles’s famously difficult proof strategy for Fermat’s Last Theorem (FLT), the theorem for
follows quickly from a weaker analogue of the abc conjecture.
- A generalization of FLT concerning powers that are sums of powers, called the Fermat-Catalan conjecture, also follows from abc.
- If a polynomial
with integer coefficients has at least three simple zeroes, then there are only finitely many positive integers
such that
for some integers
).
This raises the following natural question for us computational complexity theorists:
Does the abc conjecture imply any “shocks” in complexity theory—namely, resolving basic open questions that have been open for over half a century?
Ken and I are not aware of any. It does not seem to affect factoring, or complexity theory, or any main ingredients of our favorite P=NP problem. But it does have great impact on anything that concerns Diophantine questions. It is possible that connections may emerge at this level of detail.
Is The ABC Proved?
Here is a timeline of Mochizuki proof:
Mochizuki made his work public in August 2012 without any fanfare. Soon it was picked up and the mathematical community was made aware of the claim he has proven the abc conjecture. This started the quest to determine if his proof is correct.
The proof is long and complex. Workshops were held in 2015 and 2016 on it. The presentations did not lead to acceptance of Mochizuki’s ideas, and the proof remains unclear.
Enter Peter Scholze and Jakob Stix—two world experts on number theory. They visited Kyoto University for five days of discussions with Mochizuki in 2018. It did not resolve the correctness of the proof but did bring into focus where the difficulties lay.
They wrote a report Why abc is still a conjecture. It starts:
We, the authors of this note, came to the conclusion that there is no proof. We are going to explain where, in our opinion, the suggested proof has a problem, a problem so severe that in our opinion small modifications will not rescue the proof strategy.
Then, Mochizuki wrote a response of his view of why their claims were wrong: Comments On The Manuscript by Scholze-Stix. He said:
It should be stated clearly that the assertion that “these are inessential to the point we are making” is completely false! I made numerous attempts to explain this during the March discussions, and it is most unfortunate that we were ultimately unable to communicate regarding this issue.
The disagreement over the correctness remains: Other authors have pointed to the unresolved dispute between Mochizuki and Scholze over the correctness of this work as an instance in which the peer review process of mathematical journal publication has failed in its usual function of convincing the mathematical community as a whole of the validity of a result.
Explaining Math
Albert Einstein may have said:
“If you can’t explain it to a six year old, you don’t understand it yourself.”
Some attribute this instead to Richard Feynman. But whoever said it the mathematical community generally agrees with the point. In the 1962 book New Perspectives in Physics, by Louis De Broglie, states that Einstein, when discussing theories, said:
“
ought to lend themselves to as simple a description as that even a child could understand…”
I wonder if the requirement “explain it to a six year old” does not mean the six year old must understand it. Rather that when you explain it to them they listen politely. That is the requirement is they listen. What do you think?
Open Problems
Mochizuki still claims his proof. He violates the above rule: he cannot explain it to a six year old—not even a senior expert. It still has not yet been accepted as passing the peer review stage. See this for some general comments. And this for some more. It has lots of comments.
Can the abc ever be resolved? I wonder if there is some way to say suppose that the abc is true. And then prove some surprising complexity consequence holds? Perhaps violate P=NP for example, or some other result.
I would like you all to view (on Twitter @koitiluv1842) my Retweet of our NHK-TV-station’s summary of an April-2022 NHK-TV program explaining the outline and history of Prof. Shin’ichi Mochizuki’s IUT (= Inter-Universal Teichmueller theory/geometry) and abc-Conjecture and my Tweets-Thread No. 1 that commented on the problems with the TV program contradicting Prof. Shin’ichi Mochizuki’s own clarification of IUT and my recent Threads Nos. 2 and 3 that proved PA = Presburger A = Robinson A, which are all completely decidable and contradiction-/paradox-free (also by DMd [= diagonal-methods disproofs]).
(Note: Kyoto University has another Prof. Mochizuki = Takuro Mochizuki, who has unjustly received 2021 Math Breakthru Award with a 500 million-dollar prize for his DM-line, false D-module theory. Both of them are called “The Double-Mochizuki”.)
Thread No.1
https://twitter.com/koitiluv1842/status/1513410801172094977
(The NHK-tweet’s third blue-lettered URL links to the NHK’s program summary page.)
Thread No. 2
Presburger A = PA proof
https://twitter.com/koitiluv1842/status/1529345935129444352
Thread No. 3
PA = Robinson A proof:
https://twitter.com/koitiluv1842/status/1529351241117011968
Thread No.1.1
My earlier interpretation of IUT (1/17 ~ 16/17) and its later correction = NHK-program criticisms including the one in the above Thread No. 1, after my checking of the validity of the program (17/17):
https://twitter.com/koitiluv1842/status/1514103417534574600
This is not true. The most well-known counterexample is Ivan Fesenko, the PhD advisor of Caucher Birkar. There also exist some Japanese mathematicians who claim to have understood the proof.
The problem is rather that each of these groups are up to now still not able to convince members of the other one from their position (the proof is correct vs the proof has a flaw).
The above-mentioned TV-program-summary tweet tweeted by our TV station gives just that fool-proof explanation of abc-Conjecture that even kids could see. Pls do view it by all means.
“The proof is correct vs the proof has a flaw” is a misleading way to put it. If someone puts in a good-faith effort to understand the proof, and gets stuck at a very specific point, then someone who understands the proof ought to be able to provide more detail to explain that point. Scholze and Stix have made a very concrete and simple request: in their report, please point to which diagram it is whose commutativity is rescued by not explicitly identifying pi_1(X)’s. Someone who understands the proof ought to be able to answer this concrete technical question, but so far, nobody has stepped up to the plate. Instead, all Mochizuki and Fesenko have been able to provide is invective. Taylor Dupuy has complained that Scholze hasn’t conclusively demonstrated a flaw in the proof. So if Scholze is too stupid to understand the crucial Corollary 3.12, or is too emotionally invested to listen to reason, then someone should explain Corollary 3.12 to Dupuy. If Dupuy still doesn’t understand the proof and nobody can explain it to him, then something is very wrong.
“The proof is correct vs the proof has a flaw” is a misleading way to put it. If someone who is making a good faith effort to understand the proof gets stuck at a specific point, then someone who does understand the proof ought to be able to provide more detail to clarify what is going on. Scholze and Stix have posed a very specific question: In their report, specify which diagram it is whose commutativity is rescued by not explicitly identifying π1(X)’s. Mochizuki and Fesenko have responded with nothing but invective. Even if you think that Scholze is too stupid to understand or too emotionally invested to listen to reason, what about Taylor Dupuy? Dupuy isn’t convinced that Scholze’s argument is conclusive and is open to the possibility that the notorious Corollary 3.12 is a theorem. But nobody has explained the proof of Corollary 3.12 to Dupuy’s satisfaction either. When multiple experts are independently getting stuck at the exact same point and are asking for clarification, and no clarification is forthcoming, then there is something seriously wrong.
Here’s another playground for people who like to explore the relationship between additive and multiplicative properties of natural numbers.
https://oeis.org/wiki/Riffs_and_Rotes
https://oeis.org/search?q=A000792&sort=&language=&go=Search
The minimum occurences-numbers/Anzahlen of 1’s to make n with those 1’s, additions, multiplications, and parentheses:
ω(p) = p ⇔ p = 1+1+1+1+ … (= p times 1)
ω(4) = 4 ⇔ 4 = (1+1)(1+1)
ω(6) = 5 ⇔ 6 = (1+1)(1+1+1)
ω(8) = 6 ⇔ 8 = (1+1)(1+1)(1+1)
ω(9) = 6 ⇔ 9 = (1+1+1)(1+1+1)
ω(10) = 7 ⇔ 10 = (1+1)(1+1+1+1+1)
… .
The above 1. and 2. functions of m(n) and ω(n) were explained in an article by the above Mr. Shuu and Zeta Research Institute in “Suugaku Seminaa = Math Seminar” magazine, July 2019 issue (Tokyo: Nippon Hyooron-Sha, 2019), pp. 50, ff.
Wolfgang Keller’s dichotomy “the proof is correct vs the proof has a flaw” is misleading. As a general principle, if someone who makes a concerted effort to understand a proof hits a specific point of difficulty, then someone who understands the proof should be able to provide additional clarification. Scholze and Stix have posed a very specific question: In their report, just point out which diagram it is whose commutativity is rescued by not explicitly identifying π1(X)’s. Mochizuki and Fesenko have responded only with invective. If you think that Scholze is too stupid to understand or too emotionally invested to listen to reason, then what about Taylor Dupuy? Dupuy is not persuaded that Scholze has presented a fatal flaw, but Dupuy does not understand the proof of Corollary 3.12 either. Why doesn’t someone who understands the proof take the time to explain it to Dupuy? I’m sure Dupuy would be all ears.
Years ago, multiple experts all studied the proof independently and all of them got stuck at the same point (Corollary 3.12) and requested more clarification. No clarification has been forthcoming. The simplest explanation is that the proof has a gap. Otherwise, why is the response to a request for clarification always either silence or insults with no further mathematical details? The first 1.5 pages and the last 2.5 pages of the following document should be required reading for anyone commenting on this topic.
https://www.math.columbia.edu/~woit/szpirostillaconjecture.pdf