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

[philmoore]

Originally Posted by Trilo

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

Looks like 246, but the prospect for near-term improvement is diminishing:

http://michaelnielsen.org/polymath1/...between_primes

[CRGreathouse]

Originally Posted by TheMawn

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

There are three main steps in the method, and you lose efficiency at each step. A careful proof doesn't lose 'too much' efficiency. Zhang improved the first step, and Maynard later improved the second. The third step is now in a range where it's trivial to optimize to 100% efficiency (for the case of two primes).

Zhang's original proof made no attempt to get a small bound, and 70 million was what he got by straightforward methods which lost a lot of efficiency. Later attempts reduced this by improving different parts of each of the steps.

There are some ways of faking it. For example, assuming a hypothesis called EH is essentially pretending that we can get 100% efficiency on the first step.

[only_human]

The “bounded gaps between primes” Polymath project – a retrospective
30 September, 2014
http://terrytao.wordpress.com/2014/0...retrospective/

The (presumably) final article arising from the Polymath8 project has now been uploaded to the arXiv as “The “bounded gaps between primes” Polymath project – a retrospective“. This article, submitted to the Newsletter of the European Mathematical Society, consists of personal contributions from ten different participants (at varying levels of stage of career, and intensity of participation) on their own experiences with the project, and some thoughts as to what lessons to draw for any subsequent Polymath projects. (At present, I do not know of any such projects being proposed, but from recent experience I would imagine that some opportunity suitable for a Polymath approach will present itself at some point in the near future.)

This post will also serve as the latest (and probably last) of the Polymath8 threads (rolling over this previous post), to wrap up any remaining discussion about any aspect of this project.

[bdodson]

Originally Posted by only_human

The “bounded gaps between primes” Polymath project – a retrospective
30 September, 2014
http://terrytao.wordpress.com/2014/0...retrospective/

Excuse me; but was anyone else listening to Tao on the Colbert Report yesterday? I'm fairly
sure that I heard him introduce "prime cousin" as a pair p and p+4', both prime and then
(maybe) "sexy primes" for pair p and p+6, both prime. Then the Theorem ---- one of these
three collectons p, p+2; p, p+4; and p, p+6 is infinite. He went one to say, probably all
three are infinite, but that wasn't a theorem. That would include a proof that the gap is no
more than 6.

Maybe there was a conditional, that got left off of the interview? Nothing new on bounded
gaps in Google. Someone else heard this?

-Bruce

[CRGreathouse]

Originally Posted by bdodson

Then the Theorem ---- one of these
three collectons p, p+2; p, p+4; and p, p+6 is infinite. He went one to say, probably all
three are infinite, but that wasn't a theorem. That would include a proof that the gap is no
more than 6.

Maybe there was a conditional, that got left off of the interview? Nothing new on bounded
gaps in Google. Someone else heard this?

It's conditional on a generalized version of the Elliott-Halberstam conjecture. On 'just' EH the gap is at most 12, and unconditionally it's 246.

[only_human]

Originally Posted by CRGreathouse

It's conditional on a generalized version of the Elliott-Halberstam conjecture. On 'just' EH the gap is at most 12, and unconditionally it's 246.

http://terrytao.wordpress.com/2014/0...comment-437944

The Polymath8b paper is now published at http://www.resmathsci.com/content/1/1/12

[bdodson]

Originally Posted by CRGreathouse

It's conditional on a generalized version of the Elliott-Halberstam conjecture. On 'just' EH the gap is at most 12, and unconditionally it's 246.

Thanks. The formulation is different than we heard here in his three lectures last Sept. I do recall
reading about conditional gaps being 12 or perhaps 6.(and nothing possible below 6, by current
methods) at that time. Guess this current formulation is just a restatement of what gap. of 6 means ---
one of those three sets necessarily infinite. -Bruce

[only_human]

Originally Posted by bdodson

Guess this current formulation is just a restatement of what gap. of 6 means ---
one of those three sets necessarily infinite. -Bruce

I think it's stating something stronger. From the abstract to the 8b paper linked above:

As a consequence we can obtain a bound H₁ <= 246 unconditionally and H₁ <= 6 under the assumption of the generalized Elliott-Halberstam conjecture. Indeed under the latter conjecture, we show the stronger statement that for any admissible triple (h₁, h₂, h₃), there are infinitely many n for which at least two of n + h₁, n + h₂, n + h₃ are prime, [...]

[CRGreathouse]

Yes, that's stronger, but it still doesn't get you any more than "at least one of" {twin primes, cousin primes, sexy primes} being infinite. And actually all of the statements are of this sort, even the unconditional one.

[only_human]

Originally Posted by Primeinator

Professor Tao was recently on Comedy Central with Stephen Colbert. The interview is a bit funny (and very simplistic). Here is a link:

http://thecolbertreport.cc.com/video...lg/terence-tao

Also, I did not know it had been "shown" that either twin, cousin, or sexy primes were infinite (or all three).

Thanks for the link, I wanted to see this but hadn't got around to looking for it yet.

[only_human]

This one hour video by Terry Tao talks of the developments mentioned in this thread and elsewhere. Most helpfully for people trying to understand what this all means, he spends some time on sieve techniques.

Terry Tao, Ph.D. Small and Large Gaps Between the Primes (YouTube) Published on Oct 7, 2014, UCLA Department Of Mathematics


On the large gap side he mentions a constant that is also mentioned in this blog entry:
Long gaps between primes (16 December, 2014) for which he would reward progress:

; in the spirit of Erdös’ original prize on this problem, I would like to offer 10,000 USD for anyone who can show (in a refereed publication, of course) that the constant {c} here can be replaced by an arbitrarily large constant {C}.