Article: First proof that infinitely many prime numbers come in pairs

[philmoore]

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.

[CRGreathouse]

Originally Posted by TheMawn

I just wanted to point out, before anyone made the mistake, that the 12,042 or whatever thing does not in any way state that there MUST be a prime within 12,042 of another prime.

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.

[TimSorbet]

Originally Posted by ATH

Anyone with the knowledge to understand these papers think there will ever be proven a finite bound to consecutive primes?

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.

[only_human]

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

[CRGreathouse]

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 "?").

[henryzz]

These results are for twin primes. Could these results be put to work for triples or quads etc?

[CRGreathouse]

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.

[MattcAnderson]

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.

[firejuggler]

a bit of necro thread but

projected bound is now 628

[only_human]

Originally Posted by henryzz

These results are for twin primes. Could these results be put to work for triples or quads etc?

Perhaps. Back in mid-November, new life was breathed into the Polymath 8 project that is working on prime pairs. This article is helpful in describing that and in providing background: Together and Alone, Closing the Prime Gap
By: Erica Klarreich, November 19, 2013, Quanta Magazine (simonsfoundation.org)

Now, a preprint posted to arXiv.org on November 19 by James Maynard, a postdoctoral researcher working on his own at the University of Montreal, has upped the ante. Just months after Zhang announced his result, Maynard has presented an independent proof that pushes the gap down to 600. A new Polymath project is in the planning stages, to try to combine the collaboration’s techniques with Maynard’s approach to push this bound even lower.

“The community is very excited by this new progress,” Tao said.

Maynard’s approach applies not just to pairs of primes, but to triples, quadruples and larger collections of primes. He has shown that you can find bounded clusters of any chosen number of primes infinitely often as you go out along the number line. (Tao said he independently arrived at this result at about the same time as Maynard.)

So after Maynard's paper, the Polymath8 project split off a new Polymath8b project that incorporated Maynard's work too and he has been participating in the comments and the blog entries have been split off two additional times as the comments got numerous and to provide updates.

These maths guys also occasionally intriguingly comment on wider potential consequences as in this comment:

27 November, 2013 at 6:52 pm
Terence Tao

A small observation: somewhat frustratingly, the Zhang/Maynard methods do not quite seem to be able to establish Goldbach’s conjecture up to bounded error (i.e. all sufficiently large numbers are within O(1) of the sum of two primes). But one can “split the difference” between bounds on H and establishing Goldbach conjecture with bounded error. For instance, assuming Elliott-Halberstam, one can show that at least one of the following statements hold:

1. H₁ ≤ 6 (thus improving over the current bound of H₁ ≤ 12); or

2. Every sufficiently large even number lies within 6 of a sum of two primes.

To prove this, let N be a large multiple of 6, and consider the tuple n, n+2, n+6, N-n, N-n-2. One can check that this tuple is admissible in the sense that for every prime p, there is an n such that all five elements of the tuple are coprime to p. A slight modification of the proof of DHL[5,2] then shows that there lots of n for which at least two of the five elements of this tuple are prime; either these two elements are within 6 of each other, or sum to a number between N-2 and N+6.

If we can ever get DHL[4,2] (or more precisely, a variant of this assertion for the tuple n, n+2, N-n, N-n-2), we get a more appealing dichotomy of this type: either the twin prime conjecture is true, or every sufficiently multiple of six lies within 2 of a sum of two primes.

Polymath8b, III: Numerical optimisation of the variational problem, and a search for new sieves

The best lower bounds are not obtained through the asymptotic analysis, but rather through quadratic programming (extending the original method of Maynard). This has given significant numerical improvements to our best bounds (in particular lowering the H₁ bound from 600 to 330), but we have not yet been able to combine this method with the other potential improvements (enlarging the simplex, using MPZ distributional estimates, and exploiting upper bounds on two-point correlations) due to the computational difficulty involved.

[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?