ABC conjecture: Difference between revisions
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*[http://mathoverflow.net/questions/106321/mochizukis-proof-and-siegel-zeros Mochizuki’s proof and Siegel zeros], Mathoverflow, 4 Sept 2012 | *[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/ | *[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 | ||
*[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, September 10, 2012 | |||
*[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 | |||
Revision as of 09:30, 10 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]\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.
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
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
