One of the more famous unsolved questions of mathematics is the twin prime conjecture. There are many “twin prime” pairs of the form (p, p+2), where both p and p+2 are prime numbers. The pair (3, 5), for example. Or (5,7), (11,13), (17,19), (29,31), (10007,10009), or (10000139,10000141). The question is whether there are infinitely many such pairs.
A recent burst of work has seen progress towards answering this question, and a paper on arXiv.org by James Maynard (Centre de recherches mathématiques, Université de Montréal) reports that there are infinitely many pairs of primes separated by at most 600. See also the story in Wired. The underlying unsolved question might be cracked soon!