mersenneforum.org Article: First proof that infinitely many prime numbers come in pairs
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2013-12-09, 04:55   #23
only_human

"Gang aft agley"
Sep 2002

EAA16 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:
Hm = liminf(as n->infinity) (Pn+m - pn)

So the gap between prime pairs is H1.
Quote:
 The currently best known bounds on Hm are: (Maynard) Assuming the Elliott-Halberstam conjecture, H1 <= 12. (Polymath8b, tentative) H1 <= 330. Assuming Elliott-Halberstam, H2 <= 330. (Polymath8b, tentative) H2 <= 484,126}. Assuming Elliott-Halberstam, H4 <= 493,408. (Polymath8b) Hm <= exp( 3.817m ) for sufficiently large m. Assuming Elliott-Halberstam, Hm << e^2[I]m[/I] log m for sufficiently large m.

Last fiddled with by only_human on 2013-12-09 at 05:34

2013-12-10, 03:20   #24
CRGreathouse

Aug 2006

3×52×79 Posts

Quote:
 Originally Posted by c10ck3r "Quote: Originally Posted by henryzz These results are for twin primes. Could these results be put to work for triples or quads etc? 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?
No, it would not. A priori, 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.)

Last fiddled with by CRGreathouse on 2013-12-10 at 03:21

 2014-01-13, 06:48 #25 philmoore     "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 Elliot-Halberstam 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.
2014-01-13, 22:18   #26
CRGreathouse

Aug 2006

3×52×79 Posts

Quote:
 Originally Posted by philmoore 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?
Only a bit. I've skimmed most of Halberstam & Richert but I've only gone through the very beginning carefully.

 2014-01-14, 04:18 #27 Zeta-Flux     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...r-applications Getting into the terminology of sieve theory can be daunting, so good luck!
2014-01-14, 09:47   #28
Brian-E

"Brian"
Jul 2007
The Netherlands

2·23·71 Posts

Quote:
 Originally Posted by Zeta-Flux 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.
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.

 2014-02-04, 01:06 #29 Mathew     Nov 2009 2·52·7 Posts
2014-02-04, 01:42   #30
Xyzzy

"Mike"
Aug 2002

2×3×5×257 Posts

Quote:
 It's wonderful to see you back here. You've been greatly missed.

2014-02-04, 21:19   #31
henryzz
Just call me Henry

"David"
Sep 2007
Cambridge (GMT/BST)

2×2,861 Posts

Quote:
 Originally Posted by Mathew
Nice and easy to read. There are quite a few typos in there though.

 2014-05-27, 21:58 #32 Trilo     "W. Byerly" Aug 2013 1423*2^2179023-1 2·47 Posts any more news on the gaps? Is it still at 270, or has it been proven smaller?
 2014-05-27, 23:25 #33 TheMawn     May 2013 East. Always East. 110101111112 Posts Is there any Layman terms way of explaining why 70 million, why 270, or whatever? They seem so arbitrary...

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