# Primes gotta stick together

Update 14/05/2013: The seminar was successful: Zhang announced that his proof has already been refereed for the Annals, and everyone seems happy with it.

Hard Maths news now: there’s a rumour going round that Yitang (Tom) Zhang of the University of New Hampshire reckons he can prove that there are infinitely many different pairs of primes at most 70,000,000 apart.

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 $|p-q|<70,\!000,\!000$.

We await further news.

Source: Peter Woit’s post announcing the seminar: Number Theory News.