Twin Prime conjecture current status Can someone help me with a link to read about the status of the Twin Prime conjecture. I have browse on the internet and have read some articles but still I have no clue of the updated status of Twin Prime conjecture. Can someone explain to me the current status of the Twin Prime conjecture whether it is solved or it is still widely open.
 A: The wikipedia entry is reasonably coherent, and is pretty up-to-date.
The twin prime conjecture itself remains open, but there has been recent remarkable progress on slightly weaker results.  Namely, Y. Zhang proved that there is an infinite sequence of primes $p_n$ such that $p_{n+1} - p_n$ is uniformly bounded.
(This result is referred to as bounded prime gaps.)  
Zhang's result was significantly strengthened by J. Maynard.  This article of A. Granville is fairly up-to-date (it appeared in Nov. 2013), and discusses Zhang's and Maynard's work.  The paper becomes technical, but the abstract and introduction are fairly accessible, and included a lot of history, which is current all the way up to Nov. 2013.
As Granville discusses, even the best possible tightening of Zhang's and Maynard's arguments wouldn't improve the bound on prime gaps beyond $12$ (as far as anyone knows), while twin primes states that this bound should be $2$.  So it seems that,
despite this amazing progress, the twin primes conjecture is still quite open.
