Number theorists moved a step closer to the resolving the twin prime conjecture this week when a new paper appeared on the internet, see the AIM preprint, which gives a proof that the spacing between consecutive primes is sometimes very much smaller than the average spacing. This result was originally reported two years ago by Dan Goldston and Cem Yildirim (see the original AIM press release and technical description) but was later retracted. Now, Janos Pintz has joined the team and completed the proof of this important result. Amazingly, the new proof can be given with full details in about 8 pages. Moreover, the techniques used are familiar to number theorists. The earlier version involved some new methods which turned out to be incorrect.
There is a belief among some number theorists that a psychological barrier has been broken and that a proof of the twin prime conjecture may not be far away. Indeed, Goldston expressed such a belief during a presentation of this new work at AIM on May 24.
The precise statement of the new theorem is that for any positive number ε there exist primes p and p' such that the difference between p and p' is smaller than ε log p. The proof of an even stronger statement, namely that the difference can be as small as (log p)1/2 (log log p)2 appears in a manuscript by the three authors that has been privately circulating; the details of this more complicated proof have not been carefully scrutinized by a large audience as of yet.
Dan Goldston and Cem Yildirim have smashed all previous records on the size of small gaps between prime numbers. This work is a major step toward the centuries-old problem of showing that there are infinitely many "twin primes": prime numbers which differ by 2, such as 11 and 13, 17 and 19, 29 and 31,...
Goldston lectured on his result at AIM in Palo Alto on the evening of Friday, March 28.
On April 23rd, Andrew Granville of the Universite de Montreal and K. Soundararajan of the University of Michigan found a technical difficulty buried in one of the arguments in the preprint of Goldston and Yildrim. The main issue is that some quantities which were believed to be small error terms are actually the same order of magnitude as the main term. For now this difficulty remains unresolved.
You may enjoy reading Keith Devlin's comments on the process of producing new mathematical results and verifying their correctness.