Two prime numbers separated by 2. For instance, 17 and 19 are twin primes. It is an open question if infinitely many twin primes exist.

Note that Euclid states a proof that infinitely many primes exist. It is also open to determine if there are infinitely many pairs of primes separated by other even numbers.

Mathematicians Dan Goldston and Cem Yildirim announced an important new result in late March 2003, which while not proving that there are infinitely many twin primes does substantially advance the understanding of "Small Gaps Between Primes", as their paper is called. It is said* to be one of the most important advances in prime theory for many decades.

On his homepage Goldston describes his current research as follows:

I have been working since 1985 on methods for proving that there are arbitrarily large primes that are unusually close together. The goal is to prove that for p and p' primes the infimum of the ratio (p-p')/ log p is zero. Here log p is the average distance between primes around p, and therefore we are trying to find consecutive primes within any fixed proportion of the average spacing. The current best result of Maier from 1986 shows that this ratio is infinitely often less than 0.248. Cem Yildirim and I are currently writting a series of papers on Higher Correlations of Short Divisor Sums which we hope will provide new tools both for small gaps between primes and other problems involving primes.

Early work by Hardy and Littlewood in the 1920s proved that if the General Riemann Hypothesis is true, then p' - p is infinitely often less than (2/3) log p. Later authors removed the dependence on unproved hypotheses and reduced the constant 2/3 to 0.248. What Goldston and Yildirim have now shown is that that for any ε > 0, there are infinitely many p with p' - p < ε log p.

Goldston's page: www.math.sjsu.edu/~goldston/publications.htm
Technical description: http://aimath.org/goldston_tech
Almost contentless press release of same: http://www.aimath.org/release_goldston.html
Earlier studies: http://www.cst.cmich.edu/units/mth/weekevents.htm

* though not by either of our experts ariels and Noether, who are both a bit dubious about the "importance".