### John Baez

Shared publicly  -

Besides the Odd Goldbach Conjecture, here's another big number theory result, apparently on the same day!  Not the Twin Prime Conjecture, but a step in that direction.

In 2004, Daniel Goldston, János Pintz and Cem Yıldırım,   were able to show that there are infinitely many pairs of primes at most 16 apart... if something called the Elliott–Halberstam conjecture is true.

Yitang Zhang now claims to have shown there are infinitely many pairs of primes at most 70 million apart, without relying on any unproved conjectures.

For more, see:

http://en.wikipedia.org/wiki/Elliott%E2%80%93Halberstam_conjecture

and this nice expository article:

• K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım, http://www.ams.org/journals/bull/2007-44-01/S0273-0979-06-01142-6/S0273-0979-06-01142-6.pdf

#bigness

Recall that the Twin Prime Conjecture states that there are infinitely many primes p and q such that | p - q | = 2.

http://en.wikipedia.org/wiki/Twin_prime

This has been, 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 so one could try to arrive at a weaker statement, where 3 is replaced by some number N.

Conjecture(N): There are infinitely many primes p and q such that | p - q | < N.

Note that this is very non-obvious, because it may be that the prime numbers get more and more spaced out as they get larger, in the sense that the minimum distance between primes in [M,∞) grows as M grows. We do know that this spacing grows at most linearly, by Bertrand's posulate:

http://en.wikipedia.org/wiki/Bertrand's_postulate

which states that given any prime p there is another prime p' < 2p - 2. In fact, Erdős improved this to a logarithmic bound: for any prime p there is a prime p' < p + (c ln p). The constant c has been creeping down over the decades, and we now know, as of 2005, that c can be chosen to be as small as possible (see the Wikipedia page on twin primes linked above). However, this doesn't help with the Twin Prime Conjecture, which wants to know if we can push N in Conjecture(N) down to 3 (question for the experts: why is this so?).

So, according to Peter Woit, who got a group email from Yau, there is a seminar at Harvard later today (3pm local time) by Yitang (Tom) Zhang of U New Hampshire called "Bounded gaps between primes" (this isn't listed on the Harvard website).

http://www.math.columbia.edu/~woit/wordpress/?p=5865

He claims to have a proof of Conjecture(70 000 000), which would be very big news.

If anyone is there, please report back.