Since there seems to be some degree of interest on this list for this

sort of thing, I will mention that my buddy Danny Goldston together

with Cem Yildirim of Turky have proven the prime gap conjecture, which

now becomes the Goldston-Yildrim theorem. This tells us that if G(p) is

the gap between the prime p and the next prime above it, then

lim inf G(p)/log(p) = 0

This is quite a ways yet from the twin primes conjecture, which Danny

and just about everyone else thinks is true, but it is a major step

forward.