View source for ABC conjecture

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 > 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 [http://terrytao.wordpress.com/2012/09/18/the-probabilistic-heuristic-justification-of-the-abc-conjecture/ this blog post].

* [[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/200210/fea-granville.pdf It's As Easy As abc], Andrew Granville and Thomas J. Tucker, Notices of the AMS, November 2002.

==Mochizuki's proof==

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

# (IUTT-I) [http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20I.pdf Inter-universal Teichmuller Theory I: Construction of Hodge Theaters], Shinichi Mochizuki
# (IUTT-II) [http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20II.pdf Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic Evaluation], Shinichi Mochizuki
# (IUTT-III) [http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20III.pdf Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice], Shinichi Mochizuki
# (IUTT-IV) [http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations], Shinichi Mochizuki
# [http://www.kurims.kyoto-u.ac.jp/~motizuki/Panoramic%20Overview%20of%20Inter-universal%20Teichmuller%20Theory.pdf A Panoramic Overview of Inter-universal Teichmuller Theory], Shinichi Mochizuki
# [http://www.kurims.kyoto-u.ac.jp/~motizuki/Bogomolov%20from%20the%20Point%20of%20View%20of%20Inter-universal%20Teichmuller%20Theory.pdf Bogomolov's Proof of the Geometric Version of the Szpiro Conjecture from the Point of View of Inter-universal Teichmuller Theory], Shinichi Mochizuki: "Bogomolov’s proof may be thought of as a sort of useful elementary guide, or blueprint (perhaps even a sort of Rosetta stone!), for understanding substantial portions of inter-universal Teichmüller theory."

Progress reports:
# [http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTeich%20Verification%20Report%202013-12.pdf On the Verification of Inter-Universal Teichmüller theory: A process report (as of december 2013)], Shinichi Mochizuki
# [http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTeich%20Verification%20Report%202014-12.pdf On the Verification of Inter-Universal Teichmüller theory: A process report (as of december 2014)], Shinichi Mochizuki

See also [http://www.kurims.kyoto-u.ac.jp/~motizuki/A%20Brief%20Introduction%20to%20Inter-universal%20Geometry%20(Tokyo%202004-01).pdf these earlier slides] of Mochizuki on inter-universal Teichmuller theory.  The answers to [http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture this MathOverflow post] (and in particular [http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/106658#106658 Minhyong Kim's answer]) describe the philosophy behind Mochizuki's proof strategy.  Go Yamashita has a [http://www.kurims.kyoto-u.ac.jp/~motizuki/FAQ%20on%20Inter-Universality.pdf short FAQ on inter-universality], which is a concept that appears in Mochizuki's arguments, though it does not appear to be the central ingredient in these papers.

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

* (HAT) [http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Hodge-Arakelov%20Theory%20of%20Elliptic%20Curves.pdf The Hodge-Arakelov Theory of Elliptic Curves: Global Discretization of Local Hodge Theories], Shinichi Mochizuki
* (GTKS) [http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Galois-Theoretic%20Kodaira-Spencer%20Morphism%20of%20an%20Elliptic%20Curve.pdf The Galois-Theoretic Kodaira-Spencer Morphism of an Elliptic Curve], Shinichi Mochizuki
* (HAT-Survey-I) [http://www.kurims.kyoto-u.ac.jp/~motizuki/A%20Survey%20of%20the%20Hodge-Arakelov%20Theory%20of%20Elliptic%20Curves%20I.pdf A Survey of the Hodge-Arakelov Theory of Elliptic Curves I], Shinichi Mochizuki
* (HAT-Survey-II) [http://www.kurims.kyoto-u.ac.jp/~motizuki/A%20Survey%20of%20the%20Hodge-Arakelov%20Theory%20of%20Elliptic%20Curves%20II.pdf A Survey of the Hodge-Arakelov Theory of Elliptic Curves II], Shinichi Mochizuki
* (AbsTopIII) [http://www.kurims.kyoto-u.ac.jp/~motizuki/Topics%20in%20Absolute%20Anabelian%20Geometry%20III.pdf Topics in Absolute Anabelian Geometry III: Global Reconstruction Algorithms], Shinichi Mochizuki, RIMS Preprint 1626 (March 2008).
* (EtTh) [http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Etale%20Theta%20Function%20and%20its%20Frobenioid-theoretic%20Manifestations.pdf The Etale Theta Function and its Frobenioid-theoretic Manifestations], S. Mochizuki, Publ. Res. Inst. Math. Sci. 45 (2009), pp. 227-349.  (See also [http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Etale%20Theta%20Function%20and%20its%20Frobenioid-theoretic%20Manifestations%20(comments).pdf this list] of errata for the paper.)

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 papers (AbsTopIII), (EtTh) are directly cited heavily by the IUTT series of papers; the earlier papers (HAT), (GTKS) cover thematically related material but serve more as inspiration than as a source of mathematical results in the IUTT series.

The theory of (IUTT I-IV) is used to establish a Szpiro-type inequality, which is similar to [http://en.wikipedia.org/wiki/Szpiro's_conjecture 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

* (GenEll) [http://www.kurims.kyoto-u.ac.jp/~motizuki/Arithmetic%20Elliptic%20Curves%20in%20General%20Position.pdf  Arithmetic Elliptic Curves in General Position], S. Mochizuki, Arithmetic Elliptic Curves in General Position,Math. J. Okayama Univ. 52 (2010), pp. 1-28.

are used.  (Note that the published version of this paper requires some small corrections, listed [http://www.kurims.kyoto-u.ac.jp/~motizuki/Arithmetic%20Elliptic%20Curves%20in%20General%20Position%20(comments).pdf here].)  See [http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/107386#107386 this MathOverflow post of Vesselin Dimitrov] for more discussion.

Here are the remainder of [http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html Shinichi Mochizuki's papers], and here is the [http://en.wikipedia.org/wiki/Shinichi_Mochizuki 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 [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. Mochizuki has written [http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV%20(comments).pdf comments] in October 2012 to say that he hopes to post a revised version of Theorem 1.10, which were revised in 2013.

* 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].

===Workshops===
* '''9.-20. March 2015''': RIMS Joint Research Workshop: On the verification and further development of inter-universal Teichmuller theory: [http://www.kurims.kyoto-u.ac.jp/~motizuki/2015-03%20IUTeich%20Program%20(English).pdf program]
** Lecture Series by [http://www.kurims.kyoto-u.ac.jp/~gokun/myworks.html Go Yamashita] at Kyushu University ([http://www.math.kyushu-u.ac.jp/seminars/view/1373 announcement in japanese])
** Slides by Yuichiro Hoshi on [http://www.kurims.kyoto-u.ac.jp/~yuichiro/talk20150309.pdf Mono-anabelian Reconstruction of Number Fields]

* '''7.-11. December 2015''': [https://www.maths.nottingham.ac.uk/personal/ibf/files/symcor.iut.html Clay Mathematics Institute workshop on the theory of Shinichi Mochizuki], Oxford

===Survey articles===
*Ivan Fesenko, [https://www.maths.nottingham.ac.uk/personal/ibf/notesoniut.pdf Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki]

===Blogs===
*[http://sbseminar.wordpress.com/2012/06/12/abc-conjecture-rumor-2/ ABC conjecture rumor], Secret Blogging Seminar, 12 June, 2012
*[http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/ Mochizuki on ABC], Quomodocumque, Jordan Ellenberg, 3 Sept, 2012
*[http://richardelwes.co.uk/2012/09/04/as-easy-as-123/ As easy as 123…], Simple City, Richard Elwes' Blog, 4 Sept, 2012
*[https://plus.google.com/103703080789076472131/posts/j1sEGnPyiRu Timothy Gowers Google+], 4 Sept, 2012
*[https://plus.google.com/117663015413546257905/posts/Npu7xDniXMS John Baez Google+], 4 Sept 2012, see also a [https://plus.google.com/117663015413546257905/posts/2vTzJJSueRb repost]
**[https://plus.google.com/117663015413546257905/posts/hzqBCeujWEg John Baez Google+], 5 Sept, 2012
**[https://plus.google.com/117663015413546257905/posts/d1RsN4KnCUs John Baez Google+], 12 Sept, 2012, by Minhyong Kim.
*[https://plus.google.com/114134834346472219368/posts/c7LkaWV69KL Terence Tao Google+], 4 Sept, 2012
*[http://www.math.columbia.edu/~woit/wordpress/?p=5104 Proof of the abc Conjecture?], Not Even Wrong, 4 Sept, 2012
*[http://gaussianos.com/posible-demostracion-de-la-veracidad-de-la-conjetura-abc/ Posible demostración de la veracidad de la conjetura ABC], Gaussianos, 5 Sept, 2012.
*[http://bit-player.org/2012/the-abc-game The abc game], bit-player, 7 Sept, 2012
*[http://oumathclub.wordpress.com/2012/09/09/the-abc-conjecture/ The abc Conjecture], U. Oklahoma math club, 9 Sept, 2012
*[http://golem.ph.utexas.edu/category/2012/09/the_axgrothendieck_theorem_acc.html The Ax-Grothendieck Theorem According to Category Theory], The n-Category Café, 10 Sept, 2012
*[http://www.oblomovka.com/wp/2012/09/11/touch-of-the-galois/ touch of the galois], Oblomovka, 11 Sept, 2012
*[http://rjlipton.wordpress.com/2012/09/12/the-abc-conjecture-and-cryptography/ The ABC Conjecture And Cryptography], Gödel’s Lost Letter and P=NP, 12 Sept, 2012
*[http://mochizukidenial.wordpress.com/ Mochizuki Denial], 14 Sept 2012
*[http://leisureguy.wordpress.com/2012/09/16/abc-proof-opens-new-vistas-in-math/ “ABC” proof opens new vistas in math], Later On, 16 Sept, 2012
*[http://mathbabe.org/2012/11/14/the-abc-conjecture-has-not-been-proved The ABC Conjecture has not been proved], Mathbabe, 14 Nov, 2012.
*[https://plus.google.com/u/0/115831511988650789490/posts/hJQoYM2FS6g in IUTeich the theta function corresponds to the gaze of the little girl into the “small house”], lieven lebruyn Google+, 27 May 2013
*[https://plus.google.com/u/0/115831511988650789490/posts/FWU8YD6xnNY MochizukiDenial], lieven lebruyn Google+, 28 May 2013
*[http://www.quora.com/Joseph-Heavner/Posts/An-overview-of-Inter-universal-Teichm%C3%BCller-Theory-and-Shinichi-Mochizukis-proof-of-the-ABC-Conjecture-along-with-th An overview of Inter-universal Teichmüller Theory and Shinichi Mochizuki's proof of the ABC Conjecture, along with the current situation and how we can begin to understand this theory], Joseph Heavner, Quora, Aug 18 2013
*[http://www.math.columbia.edu/~woit/wordpress/?p=6514 Latest on abc], Not Even Wrong, 19 Dec 2013
*[https://plus.google.com/+RichardElwes/posts/jMVfRcnRaoV Richard Elwes, Google+], 20 Dec 2013
*[http://www.math.columbia.edu/~woit/wordpress/?p=7451 Peter Woit on Progress-Report 2014], 13 Jan 2015
*[https://plus.google.com/+AlexanderKruel/posts/dvUQWL7tDg2 Alexander Kruel Google+], 8 Sept 2015
*[https://plus.google.com/+DavidRoberts/posts/UMKqSdvf2WB David Roberts Google+], 8 Sept 2015

===2013 study of Geometry of Frobenioids===
*[https://plus.google.com/u/0/115831511988650789490/posts/Y1XVCDLWRP5 a baby Arithmetic Frobenioid], lieven lebruyn Google+, 29 May 2013
*[https://plus.google.com/u/0/115831511988650789490/posts/dx5vuxVewzN MinuteMochizuki 2 : a quadratic arithmetic Frobenioid], lieven lebruyn Google+, 31 May 2013
*[http://matrix.cmi.ua.ac.be/content/minutemochizuki-1 MinuteMochizuki 1], the bourbaki code, lieven's blog, 1 June 2013
*[http://matrix.cmi.ua.ac.be/content/minutemochizuki-2 MinuteMochizuki 2], the bourbaki code, lieven's blog, 1 June 2013
*[https://plus.google.com/115831511988650789490/posts/Y7okWptRtEW Mochizuki's menagerie of morphisms], lieven lebruyn Google+, 4 June 2013
*[https://plus.google.com/115831511988650789490/posts/aYDv916LeEi Mochizuki's categorical prime number sieve], lieven lebruyn Google+, 5 June 2013
*[https://plus.google.com/115831511988650789490/posts/4qxuDqXPgug Mochizuki's Frobenioids for the Working Category Theorist], lieven lebruyn Google+, 7 June 2013
*[http://matrix.cmi.ua.ac.be/content/minutemochizuki-3 MinuteMochizuki 3], lieven lebruyn Google+, 9 June 2013
*[https://plus.google.com/115831511988650789490/posts/SGM3gcvyoP1 my problem with Mochizuki's Frobenioid1] lieven lebruyn Google+, 11 June 2013
*[https://plus.google.com/115831511988650789490/posts/hK66h2artZc Mochizuki's Frobenioid reconstruction: the final bit] lieven lebruyn Google+, 12 July 2013

===Q & A===
*[http://mathoverflow.net/questions/852/what-is-inter-universal-geometry What is inter-universal geometry?], Mathoverflow, 17 Oct, 2009
*[http://mathoverflow.net/questions/106321/mochizukis-proof-and-siegel-zeros Mochizuki’s proof and Siegel zeros], Mathoverflow, 4 Sept, 2012
*[http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture Philosophy behind Mochizuki's work on the ABC conjecture], Mathoverflow, 7 Sept, 2012 (see also [http://meta.mathoverflow.net/discussion/1438/mochizuki-proof-of-abc the metapost] for this question)
*[http://cstheory.stackexchange.com/questions/12504/implications-of-proof-of-abc-conjecture-for-cs-theory Implications of proof of abc conjecture for cs theory],  Theoretical Computer Science Stackexchange, 11 Sept, 2012
*[http://mathoverflow.net/questions/107379/model-theoretic-content-of-mochizukis-teichmuller-theory-papers Model-theoretic content of Mochizuki’s Teichmüller theory papers], Mathoverflow, 17 Sept 2012
*[http://math.stackexchange.com/questions/199609/groupification-and-perfection-of-a-commutative-monoid Groupification and perfection of a commutative monoid], Mathematics Stackexchange, 20 Sept 2012
*[http://www.quora.com/As-of-September-2014-what-is-the-mathematical-communitys-current-understanding-of-Mochizukis-proof-of-the-abc-conjecture As of September 2014, what is the mathematical community's current understanding of Mochizuki's proof of the abc conjecture?] Quora, September 2014.
*[http://mathoverflow.net/questions/195353/what-is-a-frobenioid What is a Frobenioid?], Mathoverflow, 31 January 2015.
*[http://mathoverflow.net/questions/195841/what-is-an-%C3%A9tale-theta-function What is an étale theta function?], Mathoverflow, 07 February 2015.


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].

===News Media===
*[http://www.nature.com/news/proof-claimed-for-deep-connection-between-primes-1.11378 Proof claimed for deep connection between primes], Nature News, 10 September 2012, reprinted by Scientific American
*[http://www.newscientist.com/article/dn22256-fiendish-abc-proof-heralds-new-mathematical-universe.html Fiendish 'ABC proof' heralds new mathematical universe], New Scientist, 10 September 2012
*[http://news.yahoo.com/mathematician-claims-proof-connection-between-prime-numbers-131737044.html Mathematician Claims Proof of Connection between Prime Numbers], Yahoo News, 11 Sept 2012, reprinted by Huffington Post and MSNBC
*[http://news.sciencemag.org/sciencenow/2012/09/abc-conjecture.html ABC Proof Could Be Mathematical Jackpot], Science, 12 Sept 2012
*[http://www.nytimes.com/2012/09/18/science/possible-breakthrough-in-maths-abc-conjecture.html A Possible Breakthrough in Explaining a Mathematical Riddle], The New York Times, 17 Sept 2012
*[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://bostonglobe.com/ideas/2012/11/03/abc-proof-too-tough-even-for-mathematicians/o9bja4kwPuXhDeDb2Ana2K/story.html An ABC proof too tough even for mathematicians], Kevin Hartnett, 3 Nov 2012.
*[http://projectwordsworth.com/the-paradox-of-the-proof/ The Paradox of the Proof], Caroline Chen, 10 May 2013.
*[http://www.newscientist.com/article/dn26753-mathematicians-anger-over-his-unread-500page-proof.html  Mathematician's anger over his unread 500-page proof], Jacob Aron, 02 Jan 2015.
*[https://www.newscientist.com/article/dn28065-our-numbers-up-machines-will-do-maths-well-never-understand/ Our number’s up: Machines will do maths we’ll never understand], New Scientist, 26 August 2015
*[http://www.nature.com/news/the-biggest-mystery-in-mathematics-shinichi-mochizuki-and-the-impenetrable-proof-1.18509 The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof], Davide Castelvecchi, 07 October 2015, Nature News

===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
*[http://theconversation.edu.au/the-abc-conjecture-as-easy-as-1-2-3-or-not-10836 The abc conjecture, as easy as 1, 2, 3 ... or not ], The Conversation, Alexandru Ghitza, 26 Nov 2012.
*[http://www.sciencenews.org/view/generic/id/349199/description/A_theorem_in_limbo_shows_that_QED_is_not_the_last_word_in_a_mathematical_proof A theorem in limbo shows that QED is not the last word in a mathematical proof], March 25, 2013.

===See also===
*[http://ncatlab.org/nlab/show/inter-universal+Teichm%C3%BCller+theory inter-universal Teichmüller theory] at the nLab
*[https://en.wikipedia.org/wiki/Frobenioid Frobenioid] at Wikipedia