You may recall studying some of the many unusual properties of prime numbers in mathematics courses past, or perhaps you are studying them now. Some of the more noteworthy properties of primes which have intrigued mathematicians for centuries or for mere decades are the Goldbach Conjecture (1742) and the Twin Prime Conjecture (1994). Both are believed to be true, but their proofs have eluded mathematicians since their inception.
Recently, however, some significant progress has been made toward proving these conjectures.
The Twin Prime Conjecture states that there are infinitely many pairs of prime numbers that differ by 2 (e.g., 3 and 5, 11 and 13). Yitang Zhang (University of New Hampshire) has proven that there is an integer N that is less than 70,000,000 such that there are infinitely many pairs of prime numbers that differ by N. While 70,000,000 is a long ways from N = 2 (which is the case for twin primes), this is the first time it has proven that there is any such finite number. The paper has been accepted for publication in the Annals of Mathematics. An article about this work can be found on the website of the Simons Foundation: http://simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap.
The original Goldbach conjecture states that every even integer greater than two can be written as a sum of two primes. For example, 82 is the sum of primes 53 and 29. While this remains unproven, a similar conjecture also posed by Goldbach known as the "Odd Goldbach Conjecture" has been proven by Harald Helfgott (Ecole Normale Superieure). The odd Goldbach Conjecture states that every odd integer greater than 5 can be written as a sum of three primes. To read the proof visit: http://arxiv.org/abs/1305.2897.
Mathematician Terence Tao talks about the method of proof in his blog at http://terrytao.wordpress.com/2012/05/20/heuristic-limitations-of-the-circle-method. (Caution: read at your own risk!)