In a series of 4 highly technical papers, Shinichi Mochizuki claims to have proved the abc conjecture
. Though you may have never heard of it, it's a powerful conjecture in number theory that would have millions of consequences, like:
• Only finitely many numbers of the form n(n+11)(n+111) are perfect cubes.
• Only finitely many numbers of the form n! + 1111 are perfect squares.
• The equation x^a + y^b = z^c has only finitely many solutions if x,y,z are relatively prime positive integers and a,b,c are positive integers with 1/a+1/b+1/c < 1.
Here two positive integers are relatively prime
if the largest integer dividing both of them is 1. Also, n here is a positive integer. Also, we could replace the numbers 11, 111 and 1111 by any other integers if we wanted to!
I have no idea whether Mochizuki's proof is correct, and I couldn't understand it without a couple years of very hard work. All I can say is that he's a serious mathematician, a Japanese algebraic geometer who has been developing Grothendieck's ideas on so-called 'anabelian geometry'. The relevant papers are the 4 entitled 'Inter-Universal Teichmuller Theory' at the bottom of his webapge here:http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html
But what does the abc conjecture actually say
? It says that for every real number p > 1, there are only finite many triples (a,b,c) of relatively prime positive integers with a + b = c such that c > x^p, where x is the product of the prime factors of abc.
The Wikipedia article lists many consequences of this conjecture - I've just skimmed the surface.
Thanks to +Alexander Kruel
and +Scott Carnahan
for pointing this out. If anyone knows the latest news on this work, let me know!