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.

• Wikipedia page for the ABC conjecture

Mochizuki's proof

The paper: INTER-UNIVERSAL TEICHMULLER THEORY IV: LOG-VOLUME COMPUTATIONS AND SET-THEORETIC FOUNDATIONS, Shinichi Mochizuki, 30 August 2012

The previous papers:Shinichi Mochizuki's papers

• Wikipedia page for Shinichi Mochizuki

Blogs

• Mochizuki on ABC, Quomodocumque, Jordan Ellenberg, 3 Sept 2012
• As easy as 123…, Simple City, Richard Elwes' Blog, 4 Sept 2012
• ABC conjecture rumor, Secret Blogging Seminar, June 12, 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
• The abc game, bit-player, 7 Sept 2012
• The Ax-Grothendieck Theorem According to Category Theory, The n-Category Café, September 10, 2012

Q & A

• Mochizuki’s proof and Siegel zeros, Mathoverflow, 4 Sept 2012
• Philosophy behind Mochizuki’s work on the ABC conjecture, Mathoverflow, 7 Sept 2012
• Implications of proof of abc conjecture for cs theory, Theoretical Computer Science Stackexchange, September 11, 2012

Discussions

• 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

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