The twin prime conjecture (see here, here and here for more information) asserts that there are infinitely many primes that have a difference of 2. For example:
3 and 5 are twin primes;
5 and 7 are twin primes;
11 and 13 are twin primes;
17 and 19 are twin primes;
29 and 31 are twin primes; etc.
While most mathematicians believe the twin prime conjecture is correct, an explicit proof has not been found. Indeed, this has been one of the most popular unsolved problems in mathematics — not necessarily because it’s important, but for the curiosity that a conjecture so simply stated has eluded conquest by the world’s best mathematicians.
Still, research continues, and some major progress has been made in the past few years. (I like sharing this story with my students to convince them that not everything that can be known about mathematics has been figure out yet — a misconception encouraged by the structure of the secondary curriculum — and that research continues to this day.) Specifically, it was recently shown that, for some integer 
http://video.newyorker.com/watch/annals-of-ideas-yitang-zhang-s-discovery-2015-01-28
http://www.newyorker.com/magazine/2015/02/02/pursuit-beauty
For more on recent progress:
