ABC conjecture: Difference between revisions
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Mochizuki's proof |
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==Mochizuki's proof== | ==Mochizuki's proof== | ||
The paper: [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], | The paper: [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, 30 August 2012 | ||
The previous papers:[http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html Shinichi Mochizuki's papers] | The previous papers:[http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html Shinichi Mochizuki's papers] | ||
*[http://en.wikipedia.org/wiki/Shinichi_Mochizuki Wikipedia page for Shinichi Mochizuki] | |||
===Online response=== | ===Online response=== | ||
*[http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/ Mochizuki on ABC], | |||
*[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 | |||
*[http://sbseminar.wordpress.com/2012/06/12/abc-conjecture-rumor-2/ ABC conjecture rumor], June 12, 2012 | *[http://sbseminar.wordpress.com/2012/06/12/abc-conjecture-rumor-2/ ABC conjecture rumor], June 12, 2012 | ||
*[https://plus.google.com/103703080789076472131/posts/j1sEGnPyiRu Timothy Gowers Google+] | |||
*[https://plus.google.com/117663015413546257905/posts/Npu7xDniXMS John Baez Google+] | *[https://plus.google.com/103703080789076472131/posts/j1sEGnPyiRu Timothy Gowers Google+], 4 Sept 2012 | ||
*[https://plus.google.com/114134834346472219368/posts/c7LkaWV69KL Terence Tao Google+] | *[https://plus.google.com/117663015413546257905/posts/Npu7xDniXMS John Baez Google+], 4 Sept 2012 | ||
*[http://mathoverflow.net/questions/106321/mochizukis-proof-and-siegel-zeros Mochizuki’s proof and Siegel zeros], Mathoverflow | **[https://plus.google.com/117663015413546257905/posts/hzqBCeujWEg John Baez Google+], 5 Sept 2012 | ||
*[https://plus.google.com/114134834346472219368/posts/c7LkaWV69KL Terence Tao Google+], 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/what-is-the-underlying-vision-that-mochizuki-pursued-when-trying-to-prove-the-abc What is the underlying vision that Mochizuki pursued when trying to prove the ABC conjecture?], Mathoverflow, 7 Sept 2012 | |||
Revision as of 06:36, 7 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
- What is the underlying vision that Mochizuki pursued when trying to prove the ABC conjecture?, Mathoverflow, 7 Sept 2012
