twin prime conjecture
There's been progress on one of the biggest open problems in maths: the twin prime conjecture. 
This year has seen a flurry of results as mathematicians hunt down the elusive proof of the twin prime conjecture. Will they get their wish for Christmas this year? 
Agreeing to pay £50,000 for something worth £2 wouldn't win you any haggling competitions. In mathematics, however, a similar result can bring you international acclaim. This is the case with recent progress towards the famous twin prime conjecture. 

Can Dan Goldston and Cem Yalcin Yildrim repair the hole in their proof to make the biggest breakthrough in prime number theory for 80 years?

We know there is infinitely many primes, but are there infinitely many twin primes? 