Though my knowledge of number theory is terribly close to zero, the twin prime conjecture is so famous that the recent news of progress towards the result have managed to reach my ears. All of this is well explained here, but in brief let me just remind you what is this conjecture about.
A twin prime pair is an (ordered, let's say) pair of prime numbers which differ by 2, like the pair (3,5) for instance. Of course, with the exception of the pair (2,3), this is the closest two prime numbers can get. A little experimentation seems to indicate that there are many such pairs : (3,5), (5,7), (11,13), (17,19), (29,31) are the first ones. This makes the conjecture that there is an infinity of such pairs sound natural : this is the twin prime conjecture. On the other hand, if you know more about the prime numbers repartition, the conjecture, if true, will seem much more striking to you since the prime numbers are on the average farther and farther apart from each other.
Now on april this year, Yitang Zhang announced a proof of the following property :
(P_N) : there exists infinitely many primes that differ by at most N
where N is about 70 000 000. Of course the twin prime conjecture is that (P_2) is true, but anyway Zhang's result (which has now been checked by all the experts in the field) is a major breakthrough. Incidentally, Zhang, who is already in his fifties, was a complete unknown who worked alone and sometimes in difficult conditions.
Now enters Terry Tao and his excellent blog. I wish I had the time to talk about this blog before, but I had not, so this is the opportunity. Tao, who always does an incredible job at explaining thoroughly and pedagogically all kinds of subjects, now offers you the opportunity to witness real mathematic research being done, so to speak, right before your eyes. He launched the polymath8 project, which is a collaborative project whose goal is to optimize Zhang methods in order to lower the bound N. The results so far are impressive : they started from 70 000 000 in may, and they reached about 50 000 already by mid-june. Now the bound is down to 4680, and a paper is announced. However they apparently won't be able to get past the barrier of N=16 using the same methods. I wish I understood more about all of this. I encourage the readers who do to go to Tao's blog and keep us informed.