Mathematician announces that he's proved the ABC conjecture

Sep 12, 2012 by Bob Yirka report

(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 a+b=c and involves what are known as square free numbers; numbers that can't be divided by a 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 . 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 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.

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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

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User comments : 15

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axemaster
5 / 5 (1) Sep 12, 2012
In case anybody didn't get the hint, this is a Big Deal if he really has done it.
Guilherme22
3 / 5 (2) Sep 12, 2012
so what that means?
Tausch
1 / 5 (1) Sep 12, 2012
Bekanntheitsgrad - the positive side of information flow - with the speed of wildfire without the destruction.

We all look forward to understanding the implications (in the near future) if true.
sirchick
not rated yet Sep 12, 2012
Bekanntheitsgrad - the positive side of information flow - with the speed of wildfire without the destruction.

We all look forward to understanding the implications (in the near future) if true.


Say again but dumb it down for me to understand :P ?
Torbjorn_Larsson_OM
5 / 5 (1) Sep 12, 2012
I note that indeed the Fermat theorem would follow, not in general but for numbers large enough.

@ 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).
flashgordon
not rated yet Sep 12, 2012
Well, I've watched this Hodge conjecture video a bunch of times; it's a good place to start.

http://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!)
Deathclock
5 / 5 (1) Sep 12, 2012
I really wish I could understand this...
hemitite
4.2 / 5 (5) Sep 12, 2012
There is no royal road to mind bending mathematics.

I 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.
hemitite
5 / 5 (5) Sep 12, 2012
Elliptic not epileptic curves you moron! Oh, that was me...
Torbjorn_Larsson_OM
not rated yet Sep 13, 2012
Maybe I was hasty. But I note on another blog that Mochizuki relies heavily on Teichmüller theory. [ http://www.nature...-1.11378 ] Which may have a connection to the Hodge conjecture ("Hodge Theatres").

Using 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_OM
not rated yet Sep 13, 2012
On the contrary, I see claims that it is specifically number theory. [ http://www.lifesl...ers.html ]
Tausch
1 / 5 (1) Sep 13, 2012
@Dealthclock
Same boat.

Sheer unerschöpflich (inexhaustible)is the literature to aid and support an understanding of Perleman's research.

Not so here.
RazorsEdge
not rated yet Sep 17, 2012
"it will likely take years of serious work by many mathematicians to prove that the proof is correct."
Work 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.
Lex Talonis
1 / 5 (3) Sep 17, 2012
I worked all of this out ages ago.

Pretty easy - between playing pac man, eating pizza, and a few beers.

Only took half an hour too.

Mostly because I couldn't find the pencil sharpener - but no mind.

duaned
5 / 5 (1) Oct 26, 2012
I have a small difficulty with the opening paragraph. It states that most people seldom use mathematics. It is used by almost everyone, though mostly at a low level. Go work in a cabinet shop or look at a DOW jones graph-mathematics is there. The other statement seems to say that mathematics obscures the underlying concepts. I think that it illuminates the underlying concepts.

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