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

Online response

• Mochizuki on ABC, Quomodocumque, Jordan Ellenberg, 3 Sept 2012
• As easy as 123…, Simple City, Richard Elwes' Blog, 4 Sept 2012
• ABC conjecture rumor, June 12, 2012

• Timothy Gowers Google+, 4 Sept 2012
• John Baez Google+, 4 Sept 2012
  • John Baez Google+, 5 Sept 2012
• Terence Tao Google+, 4 Sept 2012

• Mochizuki’s proof and Siegel zeros, Mathoverflow, 4 Sept 2012
• Philosophy behind Mochizuki’s work on the ABC conjecture, Mathoverflow, 7 Sept 2012
• The Ax-Grothendieck Theorem According to Category Theory, The n-Category Café, September 10, 2012
• Proof claimed for deep connection between primes, Nature News, 10 September 2012
• Fiendish 'ABC proof' heralds new mathematical universe, New Scientist, 10 September 2012