ABC conjecture: Difference between revisions
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*[http://en.wikipedia.org/wiki/Shinichi_Mochizuki Wikipedia page for Shinichi Mochizuki] | *[http://en.wikipedia.org/wiki/Shinichi_Mochizuki Wikipedia page for Shinichi Mochizuki] | ||
===Blogs | ===Blogs=== | ||
*[http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/ Mochizuki on ABC], Quomodocumque, Jordan Ellenberg, 3 Sept 2012 | *[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://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], Secret Blogging Seminar, June 12, 2012 | *[http://sbseminar.wordpress.com/2012/06/12/abc-conjecture-rumor-2/ ABC conjecture rumor], Secret Blogging Seminar, June 12, 2012 | ||
*[https://plus.google.com/103703080789076472131/posts/j1sEGnPyiRu Timothy Gowers Google+], 4 Sept 2012 | *[https://plus.google.com/103703080789076472131/posts/j1sEGnPyiRu Timothy Gowers Google+], 4 Sept 2012 | ||
*[https://plus.google.com/117663015413546257905/posts/Npu7xDniXMS John Baez Google+], 4 Sept 2012 | *[https://plus.google.com/117663015413546257905/posts/Npu7xDniXMS John Baez Google+], 4 Sept 2012 | ||
**[https://plus.google.com/117663015413546257905/posts/hzqBCeujWEg John Baez Google+], 5 Sept 2012 | **[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 | *[https://plus.google.com/114134834346472219368/posts/c7LkaWV69KL Terence Tao Google+], 4 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 | |||
===Q & A=== | |||
*[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/philosophy-behind-mochizukis-work-on-the-abc-conjecture Philosophy behind Mochizuki’s work on the ABC conjecture], Mathoverflow, 7 Sept 2012 | *[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://cstheory.stackexchange.com/questions/12504/implications-of-proof-of-abc-conjecture-for-cs-theory Implications of proof of abc conjecture for cs theory], Theoretical Computer Science Stackexchange, September 11, 2012 | |||
*[http:// | ===Discussions=== | ||
*[http://news.ycombinator.com/item?id=4476367 Shin Mochizuki has released his long-rumored proof of the ABC conjecture ], Hacker News, 5 Sept 2012 | |||
**[http://news.ycombinator.com/item?id=4502856 Proof Claimed for Deep Connection between Prime Numbers], Hacker News, 11 Sept 212 | |||
*[http://science.slashdot.org/story/12/09/10/226217/possible-proof-of-abc-conjecture Possible Proof of ABC Conjecture], Slashdot, September 10, 2012 | *[http://science.slashdot.org/story/12/09/10/226217/possible-proof-of-abc-conjecture Possible Proof of ABC Conjecture], Slashdot, September 10, 2012 | ||
===News Media=== | ===News Media=== | ||
*[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 | *[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 | ||
*[http://www.newscientist.com/article/dn22256-fiendish-abc-proof-heralds-new-mathematical-universe.html Fiendish 'ABC proof' heralds new mathematical universe], New Scientist, 10 September 2012 | *[http://www.newscientist.com/article/dn22256-fiendish-abc-proof-heralds-new-mathematical-universe.html Fiendish 'ABC proof' heralds new mathematical universe], New Scientist, 10 September 2012 | ||
Revision as of 12:14, 11 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
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
- John Baez Google+, 5 Sept 2012
- Terence Tao Google+, 4 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
- Fiendish 'ABC proof' heralds new mathematical universe, New Scientist, 10 September 2012
