20130622, 17:37  #12 
"Phil"
Sep 2002
Tracktown, U.S.A.
1,117 Posts 
The Prime Number Theorem implies that the average gap between primes is ln n, so obviously, gaps can be arbitrarily large. (This was known even before the Prime Number Theorem, since n!+k is composite for k = 2, 3, ..., n.) Now, we would expect the probability of two numbers n and n+2 to both be prime to be on the order of (ln n)^{2}. The sum of 1/(ln n)^{2} diverges, so we still would expect the gap of 2 to occur infinitely often, even though the average gap becomes much larger as n increases.

20130623, 01:52  #13 
Aug 2006
1725_{16} Posts 
Right. What's more, it hasn't been proven (though it's surely true!) that infinitely often there is a pair (n, n + 12042) where both members are prime. All that is known is that there is some 2 <= k <= 12042 such that infinitely often there is a pair (n, n + k) with both members prime.

20130623, 03:51  #14 
Account Deleted
"Tim Sorbera"
Aug 2006
San Antonio, TX USA
17·251 Posts 
Assuming you mean something like "is there an integer x for which there will never be more than x consecutive composites?", nope, here's why: (x+1)#+1 is followed by at least x+1 consecutive composites. Now, there is such a thing known that for sufficiently large n, there's always a prime between n and 2n. If you mean "can that be further refined?", probably so.
Last fiddled with by MiniGeek on 20130623 at 03:52 
20130714, 00:22  #15 
"Gang aft agley"
Sep 2002
2·1,877 Posts 
Nice summation in Discover magazine; it has historical background, mentions Zhang's work, the Polymath8 project, Terence Tao and even GIMPS.
Primal Madness: Mathematicians' hunt for Twin Prime Numbers July 10, 2013 
20130714, 17:13  #16 
Aug 2006
3·5^{2}·79 Posts 
I put together a summary of the progress in improving Zhang's original bound of 70 million. The gap is on a logarithmic scale from 70 million to 18, which is the expected limit of this method.
I also labeled some improvements with notes; I chose the ones which appeared to lead to significant changes. Consider them as a lay (mis)understanding of a technical summary. I tried to include all verified bounds but the most recent ones are unverified (marked with "?"). 
20130714, 19:27  #17 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
2·2,861 Posts 
These results are for twin primes. Could these results be put to work for triples or quads etc?

20130714, 20:00  #18 
Aug 2006
1725_{16} Posts 

20130715, 11:36  #19 
"Matthew Anderson"
Dec 2010
Oregon, USA
256_{16} Posts 
It is my understanding that Zhang's work indicates that ko touples of a certain width Can be proven to contain at least two primes an infinite number of times.
Thanks to CRGreathouse for tabulating the latest, unconfirmed width. A table of narrow admissible sets can be found here Sites.google.com/site/anthonydforbes/ktpatt.txt Also, Thomas Englesma has recently made public some patterns for larger widths. 
20131106, 14:42  #20 
Apr 2010
Over the rainbow
974_{16} Posts 
a bit of necro thread but
projected bound is now 628 
20131209, 03:20  #21  
"Gang aft agley"
Sep 2002
2×1,877 Posts 
Quote:
By: Erica Klarreich, November 19, 2013, Quanta Magazine (simonsfoundation.org) Quote:
These maths guys also occasionally intriguingly comment on wider potential consequences as in this comment: Quote:
Quote:
Last fiddled with by only_human on 20131209 at 03:38 Reason: added bold emphasis to magazine quote to address henryzz's inquiry. 

20131209, 04:43  #22 
Aug 2010
Kansas
547 Posts 
"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? Last fiddled with by c10ck3r on 20131209 at 04:44 Reason: Remove extra button thingy from crappy copypaste job 
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