There is no publicly available preprint on this yet, and my information is all either second- or third-hand, but my understanding is that Zhang has managed to find a specialised improvement of the Bombieri-Vinogradov theorem (in the spirit of some famous papers of Fouvry-Iwaniec and Bombieri-Friedlander-Iwaniec, see e.g. the introduction to this recent paper for a summary) which, when combined with the method of Goldston, Pintz, and Yildirim, establishes bounded gaps between consecutive primes infinitely often.  (The original Goldston-Pintz-Yildirim paper already noted that certain types of improvement to the Bombieri-Vinogradov theorem would give such a conclusion; I do not know if Zhang's argument establishes such improvements exactly, or establishes some variant result of this type.)

I hear that some very credible experts have already refereed the paper carefully, but it may still take some time to get "official" confirmation of the correctness of the argument, particularly in the absence of a preprint.
Recall that the Twin Prime Conjecture states that there are infinitely many primes p and q such that | p - q | = 2.

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:'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).

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

If anyone is there, please report back.

#mathematics   #primenumbers   #Plus1Experiment  
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