Game of primes ends as mathematics gets too difficult - physics-math - 23 April 2014 - New Scientist:
The twin prime conjecture posits that there are an infinite number of pairs of primes separated by two, such as 3 and 5. In May 2013, Yitang Zhang at the University of New Hampshire in Durham made the first major progress by proving that there are an infinite number of prime pairs separated by at most 70 million.
Since then, researchers have been advancing his methods to find prime pairs separated by smaller intervals...
In a few weeks they had reduced the gap to just under 5 million, and today it stands at 246. In theory, the methods used could find gaps as small as six, but the researchers say they have reached a point of diminishing returns.