# ABC conjecture

(Difference between revisions)
 Revision as of 22:46, 10 September 2012 (view source) (→Online response)← Older edit Revision as of 22:48, 10 September 2012 (view source) (→Online response)Newer edit → Line 13: Line 13: *[http://en.wikipedia.org/wiki/Shinichi_Mochizuki Wikipedia page for Shinichi Mochizuki] *[http://en.wikipedia.org/wiki/Shinichi_Mochizuki Wikipedia page for Shinichi Mochizuki] - ===Online response=== + ===Blogs and forums=== Line 28: Line 28: *[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://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://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 + + ===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 22:48, 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 $c^{1-\varepsilon}$ for any fixed $\varepsilon > 0$ (if a,b,c are smooth).

This shows for instance that $(1-\varepsilon) \log N / 3$-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