The twin primes conjecture states that there are infinitely numbers n such that n and n+2 are prime. David Roberts on Google+ refers to this as “to put it mildly, EXTREMELY HARD to prove”. An equivalent statement is that there are infinitely many primes p and q such that |p−q|<3, and this, says David, allows the production of weaker conjectures:
Conjecture(N): there are infinitely many primes p and q such that |p−q|<N.
Anyway, according to a blog post by Peter Woit of Columbia University, who apparently got an email announcing it, there is a seminar today at Harvard at 3pm local time, in which “Yitang Zhang will present new results on ‘Bounded gaps between primes’”. Peter says that Zhang claims a proof of Conjecture(70,000,000), that is: there are infinitely many primes p and q such that |p−q|<70,000,000.