# ABC conjecture

(Difference between revisions)
 Revision as of 22:45, 12 September 2012 (view source) (→Blogs)← Older edit Revision as of 03:05, 13 September 2012 (view source) (→Blogs)Newer edit → Line 26: Line 26: *[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 *[http://www.oblomovka.com/wp/2012/09/11/touch-of-the-galois/ touch of the galois], Oblomovka, 11 Sept 2012 *[http://www.oblomovka.com/wp/2012/09/11/touch-of-the-galois/ touch of the galois], Oblomovka, 11 Sept 2012 + *[http://rjlipton.wordpress.com/2012/09/12/the-abc-conjecture-and-cryptography/ The ABC Conjecture And Cryptography], Gödel’s Lost Letter and P=NP, September 12, 2012 ===Q & A=== ===Q & A===

## Revision as of 03:05, 13 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