20131209, 04:55  #23  
"Gang aft agley"
Sep 2002
EAA_{16} Posts 
A correction; these results are for an infinitude of primes with small gaps between them, not twin primes (gap = 2). From Polymath8b, III: Numerical optimisation of the variational problem, and a search for new sieves, this formula:
H_{m} = liminf_{(as n>infinity)} (P_{n+m}  p_{n}) So the gap between prime pairs is H_{1}. Quote:
Last fiddled with by only_human on 20131209 at 05:34 

20131210, 03:20  #24  
Aug 2006
3×5^{2}×79 Posts 
Quote:
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 ktuples for any k. (See the last two posts.) Last fiddled with by CRGreathouse on 20131210 at 03:21 

20140113, 06:48  #25 
"Phil"
Sep 2002
Tracktown, U.S.A.
1,117 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
http://michaelnielsen.org/polymath1/...between_primes Current progress has reduced the gap to 270 unconditionally, but to 8 assuming the Generalized ElliotHalberstam conjecture. (Of course we all know 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 Sieve Methods 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. 
20140113, 22:18  #26  
Aug 2006
3×5^{2}×79 Posts 
Quote:


20140114, 04:18  #27 
May 2003
7×13×17 Posts 
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 http://www.cambridge.org/us/academic...rapplications Getting into the terminology of sieve theory can be daunting, so good luck! 
20140114, 09:47  #28 
"Brian"
Jul 2007
The Netherlands
2·23·71 Posts 
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.

20140204, 01:06  #29 
Nov 2009
2·5^{2}·7 Posts 

20140204, 01:42  #30  
"Mike"
Aug 2002
2×3×5×257 Posts 
Quote:


20140204, 21:19  #31  
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
2×2,861 Posts 
Quote:


20140527, 21:58  #32 
"W. Byerly"
Aug 2013
1423*2^21790231
2·47 Posts 
any more news on the gaps?
Is it still at 270, or has it been proven smaller?

20140527, 23:25  #33 
May 2013
East. Always East.
11010111111_{2} Posts 
Is there any Layman terms way of explaining why 70 million, why 270, or whatever? They seem so arbitrary...

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