The main objectives of the polymath8 project, initiated back in June, were to understand the recent breakthrough paper of Zhang establishing an infinite number of prime gaps bounded by a fixed constant , and then to lower that value of as much as possible. After a large number of refinements, optimisations, and other modifications to Zhang’s method, we have now lowered the value of from the initial value of down to (provisionally) , as well as to the slightly worse value of if one wishes to avoid any reliance on the deep theorems of Deligne on the Weil conjectures.

As has often been the case with other polymath projects, the pace has settled down subtantially after the initial frenzy of activity; in particular, the values of (and other key parameters, such as , , and ) have stabilised over the last few weeks. While there may still be a few small improvements in these parameters that can be wrung out of our methods, I think it is safe to say that we have cleared out most of the “low-hanging fruit” (and even some of the “medium-hanging fruit”), which means that it is time to transition to the next phase of the polymath project, namely the writing phase.

After some discussion at the previous post, we have tentatively decided on writing a single research paper, which contains (in a reasonably self-contained fashion) the details of the strongest result we have (i.e. bounded gaps with ), together with some variants, such as the bound that one can obtain without invoking Deligne’s theorems. We can of course also include some discussion as to where further improvements could conceivably arise from these methods, although even if one assumes the most optimistic estimates regarding distribution of the primes, we still do not have any way to get past the barrier of identified as the limit of this method by Goldston, Pintz, and Yildirim. This research paper does not necessarily represent the only output of the polymath8 project; for instance, as part of the polymath8 project the admissible tuples page was created, which is a repository of narrow prime tuples which can automatically accept (and verify) new submissions. (At an early stage of the project, it was suggested that we set up a computing challenge for mathematically inclined programmers to try to find the narrowest prime tuples of a given width; it might be worth revisiting this idea now that our value of has stabilised and the prime tuples page is up and running.) Other potential outputs include additional expository articles, lecture notes, or perhaps the details of a “minimal proof” of bounded gaps between primes that gives a lousy value of but with as short and conceptual a proof as possible. But it seems to me that these projects do not need to proceed via the traditional research paper route (perhaps ending up on the blog, on the wiki, or on the admissible tuples page instead). Also, these projects might also benefit from the passage of time to lend a bit of perspective and depth, especially given that there are likely to be further advances in this field from outside of the polymath project.

I have taken the liberty of setting up a Dropbox folder containing a skeletal outline of a possible research paper, and anyone who is interested in making significant contributions to the writeup of the paper can contact me to be given write access to that folder. However, I am not firmly wedded to the organisational structure of that paper, and at this stage it is quite easy to move sections around if this would lead to a more readable or more logically organised paper.

I have tried to structure the paper so that the deepest arguments – the ones which rely on Deligne’s theorems – are placed at the end of the paper, so that a reader who wishes to read and understand a proof of bounded gaps that does not rely on Deligne’s theorems can stop reading about halfway through the paper. I have also moved the top-level structure of the argument (deducing bounded gaps from a Dickson-Hardy-Littlewood claim , which in turn is established from a Motohashi-Pintz-Zhang distribution estimate , which is in turn deduced from Type I, Type II, and Type III estimates) to the front of the paper.

Of course, any feedback on the draft paper is encouraged, even from (or especially from!) readers who have been following this project on a casual basis, as this would be valuable in making sure that the paper is written in as accessible as fashion as possible. (Sometimes it is possible to be so close to a project that one loses some sense of perspective, and does not realise that what one is writing might not necessarily be as clear to other mathematicians as it is to the author.)

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1 September, 2013 at 1:55 pm

xfxieSeems 630 also survives.

http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7_630_o.mpl

1 September, 2013 at 2:21 pm

xfxieIncorrect. Based on the typo found by Gergely.

1 September, 2013 at 2:55 pm

Eytan PaldiIt is easy to see that the J-th Taylor term for is maximal for and decreasing afterwards where the remainder term can easily be estimated analytically.

2 September, 2013 at 9:51 am

Terence TaoI have created a subpage of the polymath wiki at

http://michaelnielsen.org/polymath1/index.php?title=Polymath8_grant_acknowledgments

to hold the contact information and grant acknowledgments of the Polymath8 project participants.

Of course it is not well defined exactly what a “participant” of the project is, since the level of contribution can vary from being responsible for a significant chunk of the research and writeup to leaving a single blog comment. In the past (and specifically for Polymath1 and Polymath4), what we have done is relied on self-reporting: people who feel that they have made a significant mathematical contribution to the project (of the level commensurate with that of a co-author in a traditional mathematical research paper) can add their own name, contact information, and grant information to the first section of the wiki page. Those who feel that they have made an auxiliary contribution to the project (e.g. stylistic suggestions, locating references, etc. – commensurate with a mention in the Acknowledgments section of a mathematical research paper) – can add their name to the second section of the wiki page. The dividing line between the two categories is necessarily a bit blurry, but in practice it seems that most participants are able to categorise their contributions appropriately.

Because of spam reasons, new user accounts are not currently enabled on the polymath wiki except through emailing the administrator; if you are unable to add your name to the wiki because of this, email me and I can add your name manually.

p.s. I plan to roll over this thread to a new one soon, as the number of comments here is getting quite large. We are soon reaching the point where we have draft text for each of the sections of the paper, so we can shortly begin focusing on the proofreading and polishing phases of the writeup.

2 September, 2013 at 10:52 am

Eytan PaldiIn the line below (4.53) there is a (plausible – but still unjustified!) claim that a certain sequence is decreasing.

2 September, 2013 at 2:32 pm

Polymath8: Writing the paper, II | What's new[…] main purpose of this post is to roll over the discussion from the previous Polymath8 thread, which has become rather full with comments. As with the previous thread, the main focus on […]

2 September, 2013 at 2:38 pm

Terence TaoI’m rolling over the discussion to a new thread. Of course, everyone is welcome to continue any active discussions here if it would not make sense to break up the comment thread, but any fresh discussion should probably move over to the new thread where it would be easier to follow.

21 October, 2013 at 9:05 am

cainiaozrReblogged this on ZHANG RONG.

1 May, 2014 at 12:56 pm

marouane rhaflihi,

to solve the twin prime conjecture and to find an algorithm that can find all the consecutive prime numbers you must see the prime number theorem differently , personnaly i see that this theorem is based only on estimations and there is anything exact at 100% , this is what i did and start over to understand the partition of prime in the composite numbers and i developped an algorithm to find them exactly , i also developped Pi(x) that can output the exact numbers of primes into an interval I also proved mathematically the twin prime conjecture and goldbach conjecture , you see this on my blog

http://rhaflimarouane.wordpress.com/

31 July, 2014 at 8:33 am

Together and Alone, Closing the Prime Gap | For a better Vietnam[…] Polymath project has focused lately on writing up its findings in a paper, already over 150 pages, which it has been invited to submit to the journal Algebra & Number […]

9 September, 2014 at 8:36 pm

Terry Tao’s blog and the twin prime conjecture | Actumaths[…] : they started from 70 000 000 in may, and they reached about 50 000 already by mid-june. Now the bound is down to 4680, and a paper is announced. However they apparently won't be able to get past the barrier of N=16 […]

17 September, 2014 at 5:13 am

Mathematicians Team Up on Twin Primes Conjecture | Simons Foundation | fit2read[…] Polymath project has focused lately onwriting up its findings in a paper, already over 150 pages, which it has been invited to submit to the journal Algebra & Number […]