[发一个单独的贴子以便链接到微博]

关于张益唐（Yitang Zhang）有可能在Twin Prime Conjecture方面取得了重大进展的消息，Terence Tao 在博客上发布了一个短贴：

Terence Tao, May 13, 2013 (edited) - Public

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 http://arxiv.org/abs/1108.0439 for a summary) which, when combined with the method of Goldston, Pintz, and Yildirim http://arxiv.org/abs/math.NT/0508185, 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.

原始网址：
https://plus.google.com/114134834346472219368/posts/XESxA9bL5um

可以更新你的的关于孪生素数的文章了。。 真是有点兴奋， 看到近期关于素数的突破。。

他又comment了一下，带有一点搞笑的意思
plus.google.com/u/0/114134834346472219368/posts/39tuzQ8npYt
把七千万降到了5947.064万。

虽是调侃，但从语气上看Tao已读过并认同了Zhang的证明。

把Tao的新评论复制在这里以便链接到微博：

by Terence Tao (May 30 12:03 PM):

In the last week there have been a number of "cheap" improvements to Zhang's recent result that there are infinitely many pairs of primes of distance at most 70,000,000 apart; by modifying the "easy" part of his argument, this bound has been reduced to 63.374.611 (Lewko), 59.874.594 (Trudgian), and now 59.470.640 (Morrison).

Basically, what Zhang really shows is that if H is any set of 3,500,000 integers with the property that H avoids at least one residue class mod p for each prime p, then there are infinitely many translates of H that contain at least two primes, and hence there are infinitely many pairs of primes of distance at most diam(H) apart; all the above "easy" improvements come from being a little more clever as to how to select H. If you want to get a brief chance to claim the "world record" for the best bound on small prime gaps, this could be an opportunity. (Of course, the ultimately more interesting challenge is to improve the 3,500,000 in Zhang's result, preferably with some powerful new ideas...)

by Terence Tao (May 30 1:39 PM):

I just posted a comment there in which I get an improvement to 58,885,998 conditional on a plausible claim, and with a bit of numerical calculation one should be able to obtain unconditional improvements on this bound, but I'll leave it to someone else to jump in.

Tao引用的一些改进结果出现在MathOverflow和ArXiv上，Tao的调侃也许跟这篇文章的题目有关？

http://arxiv.org/abs/1305.6369

A poor man's improvement on Zhang's result: there are infinitely many prime gaps less than 60 million

接微博网友“密报”，Tao贴了一篇长文：

http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/

http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/

http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/

奇怪似乎google reader和feedly都没抓到这篇文章。

这两天，以Tao为首的一群人对张益唐的结果做不停的改进。这种合作模式真的很值得赞赏。