The abc conjecture states that for integers a+b=c, the ratio of sqp(abc)^r/c always has some minimum value greater than zero for any value of r greater than 1 (A good basic explanation of the conjecture in the Nature news article and also here.).
Proposed independently by David Masser and Joseph Oesterle in 1985, the conjecture might not be as familiar to the wider world as Fermat’s Last Theorem, but in some ways it is more significant. It would reduce the proof for Fermat's Last Theorem, for example, to a page.
Now mathematician Shinichi Mochizuki of Kyoto University in Japan has released a 500-page proof.
According to mathematicians, if the proof is correct it will be "will be one of the most astounding achievements of mathematics of the twenty-first century.” But checking Mochizuki's proof is going to take some time-it is spread across four long papers (Inter-universal Teichmuller Theory I-IV number theory fans!) , each of which rests on earlier long papers."