September 12, 2012 report

# Mathematician announces that he's proved the ABC conjecture

(Phys.org)—In all of history there are very few names that stand out in the field of mathematics, at least among those not in the field: Euclid, Newton, Pythagoras, etc. This is likely due to several reasons, chief among them is that math is so seldom used by most people and the fact that its use in other sciences causes the underlying concepts to become overshadowed. That might change if what Shinichi Mochizuki of Kyoto University is claiming is true; that he has written a proof of the ABC conjecture. To mathematicians it's akin to the Grand Unified Theory of physics, a proof that would tie together most of the fundamental ideas in the field into one neat, fully explainable bundle.

The ABC conjecture is at its core, an association between whole numbers and is formed on the basis of the simple mathematical equation a+b=c and involves what are known as square free numbers; numbers that can't be divided by a number squared. Square free numbers are described using sqp(n) where n is the biggest such number that can be calculated by multiplying factors of n which are prime numbers. The whole idea was first proposed by two mathematicians working separately back in 1985.

While the concept of the ABC conjecture is not all that complex in and of itself, providing proof of it has proven to be impossible, until now, maybe. The proof Mochizuki came up with is 500 pages long and involves concepts that very few people understand, thus, it will likely take years of serious work by many mathematicians to prove that the proof is correct.

Anyone that has sat through higher level math classes that call for creating proofs can attest to the monumental effort that must have gone into creating such a proof, though virtually all mathematicians would agree that if the proof is indeed correct it will have been more than worth the effort. In fact, many suggest it would mark one of the most profound achievements in mathematics history, not only because of the proof itself but because of what it would mean to the science as a whole. In proving this one conjecture, many other proofs involving many other theorems would naturally follow. It would be as if Mochizuki had conceived and written proofs for hundreds of other important theorems all at once, including the famous Fermat's Last Theorem.

**More information:**

Mochizuki, S.

*Inter-universal teichmuller theory*, 4 parts:

www.kurims.kyoto-u.ac.jp/~moti … ler%20Theory%20I.pdf

www.kurims.kyoto-u.ac.jp/~moti … er%20Theory%20II.pdf

www.kurims.kyoto-u.ac.jp/~moti … r%20Theory%20III.pdf

www.kurims.kyoto-u.ac.jp/~moti … er%20Theory%20IV.pdf

© 2012 Phys.org

**Citation**: Mathematician announces that he's proved the ABC conjecture (2012, September 12) retrieved 21 October 2019 from https://phys.org/news/2012-09-mathematician-abc-conjecture.html

## User comments

axemastersirchickSay again but dumb it down for me to understand :P ?

Torbjorn_Larsson_OM@ Guilherme22:

Apparently this has recently been identified as a crucial theorem that joins a lot of work on so called Diophantine analysis together.

That analysis concerns itself with integer numbers, so have applications to discrete problems* and to computer analysis. It is also much harder and less general than analysis over real numbers.

* Such as how many balls fit into a box (discrete problem), instead of how large volume of water (continuous problem).

flashgordonhttp://www.youtub...OGo_XF2o

A teichmuller space is a universal covering of a riemann surface; so, you know that's pretty important.

At the end of the Hodge Conjecture, the audience asks how does all this relate to number theory? At which point, Tate gasps and says, "oh no, I think we need some refreshments outside!"(paraphrased anyways!)

DeathclockhemititeI actually started to read the latest proof (3d in series), and here are the fruits of my abject ignorance:

Mochizuki is attempting to do on the philosophical level is to make all the info contained in complex mathematical objects available to help with proofs outside of their hierarchical sand boxes. In other words, to "repackage" them in such a way as to make them portable to some degree.

The notation that he uses can look familiar to those of us who struggled through linear algebra back in the day, but is replete with exotic spaces full of epileptic curves "over" vast number fields along with various tensors and operators controlling and morphing through mind-numbing matrix and group comingleings, all to free one stubborn "species". So freed, it may then move "vertically" or "horizontally" through "log-theta-lattice" to some other useful destination.

After that, ABC appears to be relatively easy.

hemititeTorbjorn_Larsson_OMUsing p-adic fields, which is comparable to use analysis on reals, isn't the same as going outside of discrete problems.

And nowhere do I see the claim that this is anything but discrete analysis. Are you claiming this? (Video is > 1 h.)

Torbjorn_Larsson_OMRazorsEdgeWork will not prove the proof is correct. Work will only increase confidence in the proof (for the optimistic). I say the longer the proof the greater the chance of an error. I not interested until there is an amazing twist that produces a short proof.

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