Difference between revisions of "ABC conjecture"

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(Mochizuki's proof)
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* [[wikipedia:Abc_conjecture|Wikipedia page for the ABC conjecture]]
 
* [[wikipedia:Abc_conjecture|Wikipedia page for the ABC conjecture]]
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==Mochizuki's proof==
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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
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The previous papers:[http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html Shinichi Mochizuki's papers]
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===Online response===
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*[http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/ Mochizuki on ABC], quomodocumque.wordpress.com/2012/09/03
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*[http://sbseminar.wordpress.com/2012/06/12/abc-conjecture-rumor-2/ ABC conjecture rumor], June 12, 2012
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*[https://plus.google.com/103703080789076472131/posts/j1sEGnPyiRu Timothy Gowers Google+]
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*[https://plus.google.com/117663015413546257905/posts/Npu7xDniXMS John Baez Google+]
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*[https://plus.google.com/114134834346472219368/posts/c7LkaWV69KL Terence Tao Google+]
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*[http://mathoverflow.net/questions/106321/mochizukis-proof-and-siegel-zeros Mochizuki’s proof and Siegel zeros], Mathoverflow

Revision as of 19:35, 4 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]c^{1-\varepsilon}[/math] for any fixed [math]\varepsilon \gt 0[/math] (if a,b,c are smooth).

This shows for instance that [math](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