A correction; these results are for an infinitude of primes with small gaps between them, not twin primes (gap = 2). From [URL="http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/"]Polymath8b, III: Numerical optimisation of the variational problem, and a search for new sieves[/URL], this formula:
[I]H[/I][sub]m[/sub] = liminf[sub](as n->infinity)[/sub] ([I]P[sub]n+m[/sub][/I] - [I]p[sub]n[/sub][/I])

So the gap between prime pairs is [I]H[/I][sub]1[/sub].
[QUOTE]The currently best known bounds on [I]H[sub]m[/sub][/I] are:

(Maynard) Assuming the Elliott-Halberstam conjecture, [I]H[/I][sub]1[/sub] <= 12.
(Polymath8b, tentative) [I]H[/I][sub]1[/sub] <= 330. Assuming Elliott-Halberstam, [I]H[/I][sub]2[/sub] <= 330.
(Polymath8b, tentative) [I]H[/I][sub]2[/sub] <= 484,126}. Assuming Elliott-Halberstam, [I]H[/I][sub]4[/sub] <= 493,408.
(Polymath8b) [I]H[sub]m[/sub][/I] <= exp( 3.817[I]m[/I] ) for sufficiently large [I]m[/I]. Assuming Elliott-Halberstam, [I]H[sub]m[/sub][/I] << [I]e[/I]^[sup]2[I]m[/I][/sup] log [I]m[/I] for sufficiently large [I]m[/I].[/QUOTE]

[QUOTE=c10ck3r;361534]"Quote: Originally Posted by [B]henryzz[/B]
[I]These results are for twin primes. Could these results be put to work for triples or quads etc?[/I]

Word on the street is, no."

But if there are infinitely many prime values of p for which p+2 is also prime, wouldn't it follow that there are also (less of an infinitely) many primes p+2 for which p+6 is also prime, restricted to the case where p=2 mod 3?[/QUOTE]

No, it would not. [i]A priori[/i], there might be infinitely many primes p with p+2 prime, but all but finitely many of these primes have p+6 composite.

However since my post Maynard's fantastic work has been published, which does show a way to use Zhang's method for triples, quadruples, etc. In fact Tao has an improved version which gives (essentially) explicit bounds on how large a constant you get for k-tuples for any k. (See the last two posts.)

The blogs are interesting and lively reading, even though I understand only 1 or 2% of what they are talking about. They are referenced in
[url]http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes[/url]
Current progress has reduced the gap to 270 unconditionally, but to 8 assuming the Generalized Elliot-Halberstam conjecture. (Of course we all [I]know[/I] that the actual answer is 2.) I see that one of the four most prolific contributors to this blog is our own Pace Nielsen (aka "Zetaflux" on this forum), with some valuable analysis and computations. He may have been chased away from Mersenneforum by repeated assertions that research on odd perfect numbers is a waste of time, so I am glad that he has found a more satisfying use of his time!

My New Year's resolution is to learn more about sieve theory, so I recently purchased the reissued Dover edition of [I]Sieve Methods[/I] by Halberstam and Richert. They present in the final chapter the proof via Chen that there are an infinite number of primes p such that p+2 is either prime or a product of at most two primes. Charles Greathouse, are you conversant in this area? It seems to be a branch of number theory as intricate as algebraic number theory and analytic number theory, both of which I have studied somewhat, but I have little familiarity with sieve theory. It's always fun to learn something new.

[QUOTE=philmoore;364487]My New Year's resolution is to learn more about sieve theory, so I recently purchased the reissued Dover edition of [I]Sieve Methods[/I] by Halberstam and Richert. They present in the final chapter the proof via Chen that there are an infinite number of primes p such that p+2 is either prime or a product of at most two primes. Charles Greathouse, are you conversant in this area?[/QUOTE]

Only a bit. I've skimmed most of Halberstam & Richert but I've only gone through the very beginning carefully.

Hi philmoore. What really chased me away was that I was getting too caught up in the chess games! I still occasionally float on by to see how you all are doing.

Regarding Sieves, I strongly recommend the book by Cojocaru and Ram Murty [url]http://www.cambridge.org/us/academic/subjects/mathematics/number-theory/introduction-sieve-methods-and-their-applications[/url]

Getting into the terminology of sieve theory can be daunting, so good luck!

[QUOTE=Zeta-Flux;364535]Hi philmoore. What really chased me away was that I was getting too caught up in the chess games! I still occasionally float on by to see how you all are doing.[/QUOTE]
It's wonderful to see you back here. You've been greatly missed. I understand that the chess was, and still is, very distracting to anyone with any kind of a busy life.

David Lowry-Duda:

[URL="http://arxiv.org/abs/1401.7555"]A Friendly Intro to Sieves with a Look Towards Recent Progress on the Twin Primes Conjecture[/URL]

[QUOTE]It's wonderful to see you back here. You've been greatly missed.[/QUOTE]

:tu:

[QUOTE=Mathew;366056]David Lowry-Duda:

[URL="http://arxiv.org/abs/1401.7555"]A Friendly Intro to Sieves with a Look Towards Recent Progress on the Twin Primes Conjecture[/URL][/QUOTE]

Nice and easy to read. There are quite a few typos in there though.

any more news on the gaps?
 

Is it still at 270, or has it been proven smaller?

Is there any Layman terms way of explaining why 70 million, why 270, or whatever? They seem so arbitrary...