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Possible Proof of ABC Conjecture
submeta writes "Shinichi Mochizuki of Kyoto University has released a paper which claims to prove the decades-old ABC conjecture, which involves the relationship between prime numbers, addition, and multiplication. His solution involves thinking of numbers not as members of sets (the standard interpretation), but instead as objects which exist in 'new, conceptual universes.' As one would expect, the proof is extremely dense and difficult to understand, even for experts in the field, so it may take a while to verify. However, Mochizuki has a strong reputation, so this is likely to get attention. Proof of the conjecture could potentially lead to a revolution in number theory, including a greatly simplified proof of Fermat's Last Theorem."
Assuming the paper is correct and as impenetrable as the summary claims, this won't simplify the proof of FLT. It'd be a massive rug that the hard parts of of FLT would be swept under.
Don't do it. Ever.
Wikipedia math articles are essentially penis-measurement battles between editors who try to find the most obsucre and non-obvious manner to explain even simple arithmetic. Much like Fox News, Wikipedia math articles are bad for your brain.
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BMO
LMWTFY: http://en.wikipedia.org/wiki/Abc_conjecture#Formulations.
Obviously, that's the preliminary intuitive statement. Look further down the page for the formal statement.
Cookie Monster loves it!
Get it? ABC->Sesame St->Cookie Monster?
Oh nevermind...
From the Wikipedia article:
"The abc conjecture states that, for any > 0, there exist only finitely many triples (a, b, c) of coprime positive integers with a + b = c such that q(a, b, c) > 1 + ."
That is precisely the point of the proof, to determine under which conditions the sum of 2 integers is less than the product of the prime divisors of the 3 original numbers. I hope that is less vague :P
...and solved. I think it was the early (19)70's. A researcher named Jackson
(with the help of his brothers) came to the conclusion that it was simple as 1-2-3.
Additional verification shown that do-re-mi fit the bill as well. At the time, people
were sing all about it - I'm surprised this has come up again.
the empty set is... EMPTY!
"Rarely much smaller than"? What kind of mathematical statement is that? Are we to assume that most of the time, d is somewhat smaller than c? Are there conditions where d is larger than c? How are you supposed to get anything done with vague statements like "rarely much smaller than"?
There exists mathematical statements which sounds rather "unmathematical" at first, as an example, "almost everywhere" has a precise meaning in measure theory.
http://en.wikipedia.org/wiki/Almost_everywhere
Peter had a pretty good first glance reaction to the paper: http://www.math.columbia.edu/~woit/wordpress/?p=5104
I haven't seen any good discussions of the actual math content of the paper yet though.
Yeah, that "new, conceptual universes" line lit up my bullshit detector like a Christmas tree. But the author is well-established, so it's probably a bad translation and/or breathless hype inserted by the university PR office.
Yo dawg, I heard you like the Ackermann function, so OH GOD OH GOD OH GOD
This is one of the things I've always hated about the reporting on math, which is not only the fault of reporters but also of mathematicians.
Yes, the math is complicated, but, come on. "Conceptual universes"? That is your explanation?
And, yes, I RTFA (the first). It's pretty cool. Diophantine equations link right back to Turing/Hilbert 13 and all that jazz and the fundamental relationship between primes and everything else (you can't do math without primes, and you can't do all math without Diophantine equations). It really pleases me to see that explanation in the article.
But mathematics really needs to get less abstract in its terminology. The name needs to mean something, just like how in CS you call something "method_does_this()" instead of "method_x()".
I did not read the article yet, but my guess is that it is some journalist's lame attempt to "explain" category theory to laymen.
AccountKiller
Actually after reading a bit more, it turns out not to be as hyperbolic as it sounds. The author has come up with a whole constellation of new mathematical constructions to support his claimed proof. As the article points out, this means it'll take quite some time for mathematicians to understand these constructions before they'll be able to judge the correctness of the proof. This kind of thing would be dismissed out of hand if it came from Joe Nobody, but Shinichi Mochizuki's reputation in this case should ensure that it gets a good look. And before the crackpots hop on, no, that's not because of any ivory-tower prejudice, but simply because no sane (and busy) professional would judge that such a large personal time investment is likely be worthwhile, without some very strong past performance.
Yo dawg, I heard you like the Ackermann function, so OH GOD OH GOD OH GOD
Perhaps the Rhythm of the Primes has a new conductor.
---- Teach Peace. It's Cheaper Than War.
NBC , CBS, and FOX say about this conjecture?
This kind of thing would be dismissed out of hand if it came from Joe Nobody, but Shinichi Mochizuki's reputation in this case should ensure that it gets a good look.
I wonder how many interesting insights we miss due to such bigotry.
See http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture for a discussion on the mathematical content by experts.
Good question. Why don't you devote twenty years or so to becoming competent to judge, then spend all your time reading every crackpot's theory on trisecting angles or why pi isn't really transcendental, and let us know what you find out?
Yo dawg, I heard you like the Ackermann function, so OH GOD OH GOD OH GOD
Probably extremely few.
A friend of mine knew Shin (as he was known then) when he was an undergraduate. The guy was obscene insane-clown-level genius prodigy. Not the prodigy in the sense of the people who can shoot the lights out of the Putnam Competition but even far deeper than that, and jumping into very difficult and profound concepts by age 17 or 18. He did a small stint doing independent research with Ed Witten before moving up to pure mathematics. By 2nd or 3rd year undergrad (age 17 or so), he was already at an advanced graduate level.
I think he may be a different species.
Oh yeah and for fun he learned ancient Sanscrit.
thinking of numbers not as members of sets (the standard interpretation), but instead as objects which exist
Of course he was able to solve the problem; he used an Object Oriented framework!
Does this paper remind anyone else of chapter 9 of 'The Book' in Anathem at first glance?
"His solution involves thinking of numbers not as members of sets (the standard interpretation), but instead as objects which exist in 'new, conceptual universes.'"
huh?
You know; you could have avoided first posting your ignorance for the world to note and have just read the - linked - wikipedia article to gain some understanding.
I find these titbits about number theory absolutely fascinating... I followed a few courses at undergraduate level that touched on this material - without giving me a solid grounding. What I'd like to know is this: Is there a good textbook that would bring me up to speed with this material? I like Wikipedia articles - but I find them disjointed.. what I'd like from a textbook is something that leads me through the subject from undergraduate level onwards. Can anyone make any recommendations?
I was even right about the meaning of "almost everywhere," woot!
definition: Basically any probability 1 event, however, a probability 1 event CAN be false.
A converse example: (prob 0 event that is true) the temperature being exactly what it is currently, since temperature is continuous the probability of any exact temperature is 0 (basically because there are infinitely many other possibilities each with similar probabiliies), however, since the temperature IS, in fact, exactly what it is outside, it obviously can be true.
I think I disagree a little. I see it as unfortunate that the people expressing their frustration walked into the trap of one of the logical fallacies (which one?) of using the Four-Star laden words (NSFW denoted as ****) in doing so. However there is a point under that flawed frustrated presentation. Put in a fancier manner, the rest of Wikipedia is indeed at a generalist level, meant for people who just want to know what something is, and then go back to their life. In those cases, the "Encyclopedia is only the start" surely applies. As a random example, let's use the article on "Bioavailability", which (intuitively) means that a nutrient is useless if your body in fact cannot absorb and process it. Still at the intuitive level, it was an old criticism of vitamin pills, whereupon your body removed them before the stomach could finish peeling off the layers of nutrients all the way to the middle of the pill.
http://en.wikipedia.org/wiki/Bioavailability
That's as tricky a topic as any, but the Wiki article is in fact generalist. What the frustrated people are saying is that the content level isn't stable across all of Wikipedia. They're reacting to the wide difference in tone between that article and the math (or sometimes other engineering ones etc.)
My first Journal Entry ever, in 8 years! http://slashdot.org/journal/365947/aphelion-scifi-fantasy-horror-poetry-webzine
Nice article to spur a bit of recreational math. They even have a nice little "quality" formula to use for rating your finds. It's obvious that the place to look is powers of small numbers, especially primes.
I used a few command line tools, bc and factor, and some bash shell scripting to check a few combinations. Skimmed through the results of commands like this:
for ((i=1;i < 25;i++));do echo -n "$i "; echo "13^15-5^$i"|bc|factor;done
With that, I found a few decent quality combinations:
5 + 2^10*227^2*970060037 = 13^15, quality = 1.2417
3^28 + 2^7*5*137*4804889 = 13^12, quality = 1.1716
For sums less than 10^17, fewer than 2*10^4 combinations exist that are above a quality of 1.2. A brute force search might take a long time to find just 1.
Intellectual Property is a monopolistic, selfish, and defective concept. It is "tyranny over the mind of man"
I think it means that the number of conditions where this is true is infinite, while the number where it is not true is finite. So the ratio of true to false examples is infinite. Not sure tho whether this is a valid way of considering "almost everywhere".