Difference between revisions of "ABC conjecture"

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* [[wikipedia:Abc_conjecture|Wikipedia page for the ABC conjecture]]
 
* [[wikipedia:Abc_conjecture|Wikipedia page for the ABC conjecture]]
 +
* [http://ncatlab.org/nlab/show/abc%20conjecture nLab page for the ABC conjecture]
 
* [http://www.ams.org/notices/200002/fea-mazur.pdf Questions about Powers of Numbers], Notices of the AMS, February 2000.
 
* [http://www.ams.org/notices/200002/fea-mazur.pdf Questions about Powers of Numbers], Notices of the AMS, February 2000.
 
* [http://www.ams.org/notices/200210/fea-granville.pdf It's As Easy As abc], Andrew Granville and Thomas J. Tucker, Notices of the AMS, November 2002.
 
* [http://www.ams.org/notices/200210/fea-granville.pdf It's As Easy As abc], Andrew Granville and Thomas J. Tucker, Notices of the AMS, November 2002.
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===Specific topics===
 
===Specific topics===
  
The last part of (IUTT-IV) explores the use of different models of ZFC set theory in order to more fully develop inter-universal Teichmuller theory (this part is not needed for the applications to the abc conjecture).  There appears to be an inaccuracy in a remark in Section 3, page 43 of that paper regarding the conservative nature of the extension of ZFC by the addition of the Grothendieck universe axiom; see [http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/#comment-10605 this blog comment].  However, this remark was purely for motivational purposes and does not impact the proof of the abc conjecture.
+
* The last part of (IUTT-IV) explores the use of different models of ZFC set theory in order to more fully develop inter-universal Teichmuller theory (this part is not needed for the applications to the abc conjecture).  There appears to be an inaccuracy in a remark in Section 3, page 43 of that paper regarding the conservative nature of the extension of ZFC by the addition of the Grothendieck universe axiom; see [http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/#comment-10605 this blog comment].  However, this remark was purely for motivational purposes and does not impact the proof of the abc conjecture.
  
There is some discussion at [http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/107279#107279 this MathOverflow post] as to whether the explicit bounds for the abc conjecture are too strong to be consistent with known or conjectured lower bounds on abc.  In particular, there appears to be a serious issue with the main Diophantine inequality (Theorem 1.10 of IUTT-IV), in that it appears to be inconsistent with commonly accepted conjectures, namely the abc conjecture and the uniform Serre open image conjecture.
+
* There is some discussion at [http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/107279#107279 this MathOverflow post] as to whether the explicit bounds for the abc conjecture are too strong to be consistent with known or conjectured lower bounds on abc.  In particular, there appears to be a serious issue with the main Diophantine inequality (Theorem 1.10 of IUTT-IV), in that it appears to be inconsistent with commonly accepted conjectures, namely the abc conjecture and the uniform Serre open image conjecture.
  
The question of whether the results in this paper can be made completely effective (which would be of importance for several applications) is discussed in some of the comments to [http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/ this blog post].
+
* The question of whether the results in this paper can be made completely effective (which would be of importance for several applications) is discussed in some of the comments to [http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/ this blog post].
  
 +
* The category and topos theory viewpoint is discussed at the [http://nforum.mathforge.org/discussion/4260/abc-conjecture nForum page for the abc conjecture].
 
===Blogs===
 
===Blogs===
 
*[http://sbseminar.wordpress.com/2012/06/12/abc-conjecture-rumor-2/ ABC conjecture rumor], Secret Blogging Seminar, 12 June, 2012
 
*[http://sbseminar.wordpress.com/2012/06/12/abc-conjecture-rumor-2/ ABC conjecture rumor], Secret Blogging Seminar, 12 June, 2012
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Note that Mathoverflow has a number of policies and guidelines regarding appropriate questions and answers to post on that site; see [http://mathoverflow.net/faq this FAQ for details].
 
Note that Mathoverflow has a number of policies and guidelines regarding appropriate questions and answers to post on that site; see [http://mathoverflow.net/faq this FAQ for details].
 
===Discussions===
 
*[http://news.ycombinator.com/item?id=4476367 Shin Mochizuki has released his long-rumored proof of the ABC conjecture ], Hacker News, 5 Sept 2012
 
**[http://news.ycombinator.com/item?id=4502856 Proof Claimed for Deep Connection between Prime Numbers], Hacker News, 11 Sept 212
 
*[http://science.slashdot.org/story/12/09/10/226217/possible-proof-of-abc-conjecture Possible Proof of ABC Conjecture], Slashdot, September 10, 2012
 
  
 
===News Media===
 
===News Media===
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*[http://www.telegraph.co.uk/news/worldnews/asia/japan/9552155/Worlds-most-complex-mathematical-theory-cracked.html World's most complex mathematical theory 'cracked'], The Telegraph, 19 Sept 2012, reprinted by several other news outlets
 
*[http://www.telegraph.co.uk/news/worldnews/asia/japan/9552155/Worlds-most-complex-mathematical-theory-cracked.html World's most complex mathematical theory 'cracked'], The Telegraph, 19 Sept 2012, reprinted by several other news outlets
 
*[http://www.dailyprincetonian.com/2012/09/20/31183/ U.-educated mathematician offers proof of pivotal number theory conjecture], The Daily Princetonian, 20 Sept 2012
 
*[http://www.dailyprincetonian.com/2012/09/20/31183/ U.-educated mathematician offers proof of pivotal number theory conjecture], The Daily Princetonian, 20 Sept 2012
 +
 +
===Crowd News===
 +
*[http://news.ycombinator.com/item?id=4476367 Shin Mochizuki has released his long-rumored proof of the ABC conjecture ], Hacker News, 5 Sept 2012
 +
**[http://news.ycombinator.com/item?id=4502856 Proof Claimed for Deep Connection between Prime Numbers], Hacker News, 11 Sept 212
 +
*[http://science.slashdot.org/story/12/09/10/226217/possible-proof-of-abc-conjecture Possible Proof of ABC Conjecture], Slashdot, September 10, 2012
 +
*[http://www.metafilter.com/119847/Mathematics-world-abuzz-with-a-proof-of-the-ABC-Conjecture Mathematics world abuzz with a proof of the ABC Conjecture], MetaFilter, 11 Sept 2012

Revision as of 15:30, 27 September 2012

The abc conjecture asserts, roughly speaking, that if a+b=c and a,b,c are coprime, then a,b,c cannot all be too smooth; in particular, the product of all the primes dividing a, b, or c has to exceed [math]c^{1-\varepsilon}[/math] for any fixed [math]\varepsilon \gt 0[/math] (if a,b,c are smooth).

This shows for instance that [math](1-\varepsilon) \log N / 3[/math]-smooth a,b,c of size N which are coprime cannot sum to form a+b=c. This unfortunately seems to be too weak to be of much use for the finding primes project.

A probabilistic heuristic justification for the ABC conjecture can be found at this blog post.

Mochizuki's proof

Papers

Mochizuki's claimed proof of the abc conjecture is conducted primarily through the following series of four papers:

  1. (IUTT-I) Inter-universal Teichmuller Theory I: Construction of Hodge Theaters, Shinichi Mochizuki
  2. (IUTT-II) Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic Evaluation, Shinichi Mochizuki
  3. (IUTT-III) Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice, Shinichi Mochizuki
  4. (IUTT-IV) Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations, Shinichi Mochizuki, 30 August 2012

See also these earlier slides of Mochizuki on inter-universal Teichmuller theory. The answers to this MathOverflow post (and in particular Minhyong Kim's answer) describe the philosophy behind Mochizuki's proof strategy.

The argument also relies heavily on Mochizuki's previous work on the Hodge-Arakelov theory of elliptic curves, including the following references:

Anyone seeking to get a thorough "bottom-up" understanding of Mochizuki's argument will probably be best advised to start with these latter papers first.

The theory of (IUTT I-IV) is used to establish a Szpiro-type inequality, which is similar to Szpiro's conjecture but with an additional genericity hypothesis on a certain parameter [math]\ell[/math]. In order to then deduce the true Szpiro's conjecture (which is essentially equivalent to the abc conjecture), the results from the paper

are used. (Note that the published version of this paper requires some small corrections, listed here.) See this MathOverflow post of Vesselin Dimitrov for more discussion.

Here are the remainder of Shinichi Mochizuki's papers, and here is the Wikipedia page for Shinichi Mochizuki.

Specific topics

  • The last part of (IUTT-IV) explores the use of different models of ZFC set theory in order to more fully develop inter-universal Teichmuller theory (this part is not needed for the applications to the abc conjecture). There appears to be an inaccuracy in a remark in Section 3, page 43 of that paper regarding the conservative nature of the extension of ZFC by the addition of the Grothendieck universe axiom; see this blog comment. However, this remark was purely for motivational purposes and does not impact the proof of the abc conjecture.
  • There is some discussion at this MathOverflow post as to whether the explicit bounds for the abc conjecture are too strong to be consistent with known or conjectured lower bounds on abc. In particular, there appears to be a serious issue with the main Diophantine inequality (Theorem 1.10 of IUTT-IV), in that it appears to be inconsistent with commonly accepted conjectures, namely the abc conjecture and the uniform Serre open image conjecture.
  • The question of whether the results in this paper can be made completely effective (which would be of importance for several applications) is discussed in some of the comments to this blog post.

Blogs

Q & A

Note that Mathoverflow has a number of policies and guidelines regarding appropriate questions and answers to post on that site; see this FAQ for details.

News Media

Crowd News