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]\displaystyle{ c^{1-\varepsilon} }[/math] for any fixed [math]\displaystyle{ \varepsilon \gt 0 }[/math] (if a,b,c are smooth).

This shows for instance that [math]\displaystyle{ (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.

• Wikipedia page for the ABC conjecture
• nLab page for the ABC conjecture
• Questions about Powers of Numbers, Notices of the AMS, February 2000.
• 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 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:

• (HAT) The Hodge-Arakelov Theory of Elliptic Curves: Global Discretization of Local Hodge Theories, Shinichi Mochizuki
• (GTKS) The Galois-Theoretic Kodaira-Spencer Morphism of an Elliptic Curve, Shinichi Mochizuki
• (HAT-Survey-I) A Survey of the Hodge-Arakelov Theory of Elliptic Curves I, Shinichi Mochizuki
• (HAT-Survey-II) A Survey of the Hodge-Arakelov Theory of Elliptic Curves II, Shinichi Mochizuki
• (AbsTopIII) Topics in Absolute Anabelian Geometry III: Global Reconstruction Algorithms, Shinichi Mochizuki, RIMS Preprint 1626 (March 2008).
• (EtTh) The Etale Theta Function and its Frobenioid-theoretic Manifestations, S. Mochizuki, Publ. Res. Inst. Math. Sci. 45 (2009), pp. 227-349. (See also 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 Szpiro's conjecture but with an additional genericity hypothesis on a certain parameter [math]\displaystyle{ \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) 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 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. Mochizuki has written comments in October 2012 to say that he hopes to post a revised version of Theorem 1.10 and its proof in the not too distant future.

• 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.

• The category and topos theory viewpoint is discussed at the nForum page for the abc conjecture.

Blogs

• ABC conjecture rumor, Secret Blogging Seminar, 12 June, 2012
• Mochizuki on ABC, Quomodocumque, Jordan Ellenberg, 3 Sept, 2012
• As easy as 123…, Simple City, Richard Elwes' Blog, 4 Sept, 2012
• Timothy Gowers Google+, 4 Sept, 2012
• John Baez Google+, 4 Sept 2012, see also a repost
  • John Baez Google+, 5 Sept, 2012
  • John Baez Google+, 12 Sept, 2012, by Minhyong Kim.
• Terence Tao Google+, 4 Sept, 2012
• Proof of the abc Conjecture?, Not Even Wrong, 4 Sept, 2012
• Posible demostración de la veracidad de la conjetura ABC, Gaussianos, 5 Sept, 2012.
• The abc game, bit-player, 7 Sept, 2012
• The abc Conjecture, U. Oklahoma math club, 9 Sept, 2012
• The Ax-Grothendieck Theorem According to Category Theory, The n-Category Café, 10 Sept, 2012
• touch of the galois, Oblomovka, 11 Sept, 2012
• The ABC Conjecture And Cryptography, Gödel’s Lost Letter and P=NP, 12 Sept, 2012
• “ABC” proof opens new vistas in math, Later On, 16 Sept, 2012
• The ABC Conjecture has not been proved, Mathbabe, 14 Nov, 2012.

Q & A

• What is inter-universal geometry?, Mathoverflow, 17 Oct, 2009
• Mochizuki’s proof and Siegel zeros, Mathoverflow, 4 Sept, 2012
• Philosophy behind Mochizuki's work on the ABC conjecture, Mathoverflow, 7 Sept, 2012 (see also the metapost for this question)
• Implications of proof of abc conjecture for cs theory, Theoretical Computer Science Stackexchange, 11 Sept, 2012
• Model-theoretic content of Mochizuki’s Teichmüller theory papers, Mathoverflow, 17 Sept 2012
• Groupification and perfection of a commutative monoid, Mathematics Stackexchange, 20 Sept 2012

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

• Proof claimed for deep connection between primes, Nature News, 10 September 2012, reprinted by Scientific American
• Fiendish 'ABC proof' heralds new mathematical universe, New Scientist, 10 September 2012
• Mathematician Claims Proof of Connection between Prime Numbers, Yahoo News, 11 Sept 2012, reprinted by Huffington Post and MSNBC
• ABC Proof Could Be Mathematical Jackpot, Science, 12 Sept 2012
• A Possible Breakthrough in Explaining a Mathematical Riddle, The New York Times, 17 Sept 2012
• World's most complex mathematical theory 'cracked', The Telegraph, 19 Sept 2012, reprinted by several other news outlets
• U.-educated mathematician offers proof of pivotal number theory conjecture, The Daily Princetonian, 20 Sept 2012
• An ABC proof too tough even for mathematicians, Kevin Hartnett, 3 Nov 2012.

Crowd News

• Shin Mochizuki has released his long-rumored proof of the ABC conjecture , Hacker News, 5 Sept 2012
  • Proof Claimed for Deep Connection between Prime Numbers, Hacker News, 11 Sept 212
• Possible Proof of ABC Conjecture, Slashdot, September 10, 2012
• Mathematics world abuzz with a proof of the ABC Conjecture, MetaFilter, 11 Sept 2012
• The abc conjecture, as easy as 1, 2, 3 ... or not , The Conversation, Alexandru Ghitza, 26 Nov 2012.