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

[QUOTE]It’s a result only a mathematician could love. Researchers hoping to get ‘2’ as the answer for a long-sought proof involving pairs of prime numbers are celebrating the fact that a mathematician has wrestled the value down from infinity to 70 million.[/QUOTE]

[url]http://www.nature.com/news/first-proof-that-infinitely-many-prime-numbers-come-in-pairs-1.12989[/url]

And a related blog: [url]http://golem.ph.utexas.edu/category/2013/05/bounded_gaps_between_primes.html[/url]

new theorem proven
 

well, [url]http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_primes_conjecture_a_huge_discovery_about_prime_numbers.html[/url]

Can't say I have enough knowledge to refute or approve the proof, but it might be of some interest.

[code]
What about the gaps between consecutive primes? You might think that, because prime numbers get
rarer and rarer as numbers get bigger, that they also get farther and farther apart. On average,
that’s indeed the case. But what Yitang Zhang just proved is that there are infinitely many pairs of
primes that differ by at most 70,000,000. In other words, that the gap between one prime and
the next is bounded by 70,000,000 infinitely often—thus, the “bounded gaps” conjecture.
[/code]

And, a first look at the paper
[url]http://blogs.ethz.ch/kowalski/2013/05/21/bounded-gaps-between-primes/[/url]

It's been just over a month since Zhang's paper "Bounded gaps between primes." Since then, the Polymath8 page shows that the bounded gap may have reduced from 70,000,000 to less than 61,000.
[url]http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes[/url]

It looks like the bound has been reduced to 12,042. So there are an infinite number of prime pairs a distance of 12,042 or less apart. Exciting!

Impressive, indeed, and in only 5 week. Now it might become difficult to improve he bound.

I thought 70 million was pretty exciting but that seems to be old news.

I should note for those who may be misinterpreting this proof: It does NOT say that there is a prime after 70 million or twelve thousand or whatever numbers. What it is saying is that there are infinitely many primes with at most X in between them. It's actually a pretty weak statement.

The proof does NOT guarantee every prime has a close neighbour. If there is only a single prime number in between 10[SUP]100,000,000[/SUP] and 10[SUP]1,000,000,000[/SUP] (this is a gap of basically 10[SUP]1,000,000,000[/SUP] which is a LOT bigger than even 10[SUP]7[/SUP]) the proof still holds. It is just saying that there is ALWAYS a next set of sibling primes.

Channeling my inner RDS, what you say is a bit o gibberish.. The "statement" is quite strong, and it is a step in proving twin primes conjecture. The other two fragments about what the result "does not say" are "more than a bit" of gibberish, first because we already know that the gap between the primes can be [strike]mad[/strike] made arbitrarily large, and the second because we also already know that there is always a prime between n and 2n for any n. (of course, we understand that you used those powers of ten in a figurative sense, but still..... there is a math forum here...)

Alright. I'll give you that one. It's fairly strong in what it has set out to do but there is quite little use outside the twin primes conjecture.

All I meant to say was that it doesn't really affect the actual search for primes.

I overlooked the fact that there is a prime between n and 2n. The fact still remains that, as you said, the gap between primes is absolutely unbounded. 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.

[QUOTE=TheMawn;344094]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.[/QUOTE]I don't see where anyone suggested such a thing. I think we here all knew what the announcement meant.

[QUOTE=TheMawn;344094]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.[/QUOTE]
That is indeed very true. To our disappointment, otherwise it should be very easy for us to find primes, and get the EFF's money... :razz:, we would only have to test about 12k consecutive numbers of 100M digits, which would be most of them eliminated by as simple Erathostenes sieve, that's life...

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

It does not seem possible if the number of primes below n follows roughly n/ln n which means the average gap increases, but these proofs with infinite pairs of primes below 70,000,000 or even lower also seem counter intuitive.

The Prime Number Theorem implies that the [B]average[/B] 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)[SUP]2[/SUP]. The sum of 1/(ln n)[SUP]2[/SUP] 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.

[QUOTE=TheMawn;344094]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.[/QUOTE]

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.

[QUOTE=ATH;344135]Anyone with the knowledge to understand these papers think there will ever be proven a finite bound to consecutive primes?[/QUOTE]

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.

Nice summation in Discover magazine; it has historical background, mentions Zhang's work, the Polymath8 project, Terence Tao and even GIMPS.
[URL="http://blogs.discovermagazine.com/crux/2013/07/10/primal-madness-mathematicians-hunt-for-twin-prime-numbers/"]Primal Madness: Mathematicians' hunt for Twin Prime Numbers[/URL] July 10, 2013

I put together a [url=http://goo.gl/gzWEo]summary[/url] 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 "?").

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

[QUOTE=henryzz;346327]These results are for twin primes. Could these results be put to work for triples or quads etc?[/QUOTE]

Word on the street is, no.

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.

a bit of necro thread but
[youtube]vkMXdShDdtY[/youtube]
projected bound is now 628

[QUOTE=henryzz;346327]These results are for twin primes. Could these results be put to work for triples or quads etc?[/QUOTE]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: [URL="https://www.simonsfoundation.org/quanta/20131119-together-and-alone-closing-the-prime-gap/"]Together and Alone, Closing the Prime Gap[/URL]
By: Erica Klarreich, November 19, 2013, Quanta Magazine (simonsfoundation.org)
[QUOTE]Now, a [URL="http://arxiv.org/abs/1311.4600"]preprint posted to arXiv.org[/URL] 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.

[B]Maynard’s approach applies not just to pairs of primes, but to triples, quadruples and larger collections of primes.[/B] 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.)[/QUOTE]

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:[QUOTE]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. [I]H[/I][SUB]1[/SUB] ≤ 6 (thus improving over the current bound of [I]H[/I][SUB]1[/SUB] ≤ 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.[/QUOTE]
[URL="http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves"]Polymath8b, III: Numerical optimisation of the variational problem, and a search for new sieves[/URL]
[QUOTE]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 [I]H[/I][SUB]1[/SUB] 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.[/QUOTE]

"Quote: Originally Posted by [B]henryzz[/B]
[I]These results are for twin primes. Could these results be put to work for triples or quads etc?[/I]

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?

A correction; these results are for an infinitude of primes with small gaps between them, not twin primes (gap = 2). From [URL="http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/"]Polymath8b, III: Numerical optimisation of the variational problem, and a search for new sieves[/URL], this formula:
[I]H[/I][sub]m[/sub] = liminf[sub](as n->infinity)[/sub] ([I]P[sub]n+m[/sub][/I] - [I]p[sub]n[/sub][/I])

So the gap between prime pairs is [I]H[/I][sub]1[/sub].
[QUOTE]The currently best known bounds on [I]H[sub]m[/sub][/I] are:

(Maynard) Assuming the Elliott-Halberstam conjecture, [I]H[/I][sub]1[/sub] <= 12.
(Polymath8b, tentative) [I]H[/I][sub]1[/sub] <= 330. Assuming Elliott-Halberstam, [I]H[/I][sub]2[/sub] <= 330.
(Polymath8b, tentative) [I]H[/I][sub]2[/sub] <= 484,126}. Assuming Elliott-Halberstam, [I]H[/I][sub]4[/sub] <= 493,408.
(Polymath8b) [I]H[sub]m[/sub][/I] <= exp( 3.817[I]m[/I] ) for sufficiently large [I]m[/I]. Assuming Elliott-Halberstam, [I]H[sub]m[/sub][/I] << [I]e[/I]^[sup]2[I]m[/I][/sup] log [I]m[/I] for sufficiently large [I]m[/I].[/QUOTE]

[QUOTE=c10ck3r;361534]"Quote: Originally Posted by [B]henryzz[/B]
[I]These results are for twin primes. Could these results be put to work for triples or quads etc?[/I]

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?[/QUOTE]

No, it would not. [i]A priori[/i], 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.)

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
[url]http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes[/url]
Current progress has reduced the gap to 270 unconditionally, but to 8 assuming the Generalized Elliot-Halberstam conjecture. (Of course we all [I]know[/I] 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 [I]Sieve Methods[/I] 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.

[QUOTE=philmoore;364487]My New Year's resolution is to learn more about sieve theory, so I recently purchased the reissued Dover edition of [I]Sieve Methods[/I] 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?[/QUOTE]

Only a bit. I've skimmed most of Halberstam & Richert but I've only gone through the very beginning carefully.

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 [url]http://www.cambridge.org/us/academic/subjects/mathematics/number-theory/introduction-sieve-methods-and-their-applications[/url]

Getting into the terminology of sieve theory can be daunting, so good luck!

[QUOTE=Zeta-Flux;364535]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.[/QUOTE]
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.

David Lowry-Duda:

[URL="http://arxiv.org/abs/1401.7555"]A Friendly Intro to Sieves with a Look Towards Recent Progress on the Twin Primes Conjecture[/URL]

[QUOTE]It's wonderful to see you back here. You've been greatly missed.[/QUOTE]

:tu:

[QUOTE=Mathew;366056]David Lowry-Duda:

[URL="http://arxiv.org/abs/1401.7555"]A Friendly Intro to Sieves with a Look Towards Recent Progress on the Twin Primes Conjecture[/URL][/QUOTE]

Nice and easy to read. There are quite a few typos in there though.

any more news on the gaps?
 

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

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

[QUOTE=Trilo;374412]Is it still at 270, or has it been proven smaller?[/QUOTE]

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

[url]http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes[/url]

[QUOTE=TheMawn;374418]Is there any Layman terms way of explaining why 70 million, why 270, or whatever? They seem so arbitrary...[/QUOTE]

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.

The “bounded gaps between primes” Polymath project – a retrospective
30 September, 2014
[url]http://terrytao.wordpress.com/2014/09/30/the-bounded-gaps-between-primes-polymath-project-a-retrospective/[/url]
[QUOTE]The (presumably) final article arising from the [URL="http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes"]Polymath8 project[/URL] has now been uploaded to the arXiv as “[URL="http://arxiv.org/abs/1409.8361"]The “bounded gaps between primes” Polymath project – a retrospective[/URL]“. This article, submitted to the [URL="http://www.ems-ph.org/journals/journal.php?jrn=news"]Newsletter of the European Mathematical Society[/URL], 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 [URL="http://terrytao.wordpress.com/2014/07/20/variants-of-the-selberg-sieve-and-bounded-intervals-containing-many-primes/"]this previous post[/URL]), to wrap up any remaining discussion about any aspect of this project.[/QUOTE]

[QUOTE=only_human;384245]The “bounded gaps between primes” Polymath project – a retrospective
30 September, 2014
[url]http://terrytao.wordpress.com/2014/09/30/the-bounded-gaps-between-primes-polymath-project-a-retrospective/[/url][/QUOTE]

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

[QUOTE=bdodson;387584]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?[/QUOTE]

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.

[QUOTE=CRGreathouse;387588]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.[/QUOTE][url]http://terrytao.wordpress.com/2014/09/30/the-bounded-gaps-between-primes-polymath-project-a-retrospective/#comment-437944[/url]
[QUOTE]The Polymath8b paper is now published at [url]http://www.resmathsci.com/content/1/1/12[/url][/QUOTE]

[QUOTE=CRGreathouse;387588]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.[/QUOTE]

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

[QUOTE=bdodson;387608]Guess this current formulation is just a restatement of what gap. of 6 means ---
one of those three sets necessarily infinite. -Bruce[/QUOTE]I think it's stating something stronger. From the abstract to the 8b paper linked above:
[QUOTE]
As a consequence we can obtain a bound [I]H[/I][sub]1[/sub] <= 246 unconditionally and [I]H[/I][sub]1[/sub] <= 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 ([I]h[/I][sub]1[/sub], [I]h[/I][sub]2[/sub], [I]h[/I][sub]3[/sub]), there are infinitely many [I]n[/I] for which at least two of [I]n[/I] + [I]h[/I][sub]1[/sub], [I]n[/I] + [I]h[/I][sub]2[/sub], [I]n[/I] + [I]h[/I][sub]3[/sub] are prime, [...][/QUOTE]

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.

[QUOTE=Primeinator;387856]Professor Tao was recently on Comedy Central with Stephen Colbert. The interview is a bit funny (and very simplistic). Here is a link:

[url]http://thecolbertreport.cc.com/videos/6wtwlg/terence-tao[/url]

Also, I did not know it had been "shown" that either twin, cousin, or sexy primes were infinite (or all three).[/QUOTE]Thanks for the link, I wanted to see this but hadn't got around to looking for it yet.

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.

[URL="http://youtube.com/watch?v=pp06oGD4m00"]Terry Tao, Ph.D. Small and Large Gaps Between the Primes [/URL](YouTube) Published on Oct 7, 2014, UCLA Department Of Mathematics
[YOUTUBE]pp06oGD4m00[/YOUTUBE]

On the large gap side he mentions a constant that is also mentioned in this blog entry:
[URL="http://terrytao.wordpress.com/2014/12/16/long-gaps-between-primes/"]Long gaps between primes[/URL] (16 December, 2014) for which he would reward progress:
[QUOTE]; 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}.[/QUOTE]

It seems that everyone here pretty much expects there to be an infinite number of twin primes. However, What do people think about the possible infinitude of cousin primes, sexy primes etc.? Might there be an infinite number of primes p and p+n where n is any even number?

Along similar lines perhaps, the Green-Tao theorem states that there are arbitrarily long arithmetic progressions of primes. I could pick "Graham's number" as the length and surely one of that length must exist, but does this extend to arithmetic progressions of infinite length?

Also, does this theorem or any other give an indication to the number of sequences of length k? Might there be an infinite number of each such sequence?

Hi Math People,

The [URL="http://mathworld.wolfram.com/k-TupleConjecture.html"]k-tuple conjecture[/URL] states that there for any admissible [URL="http://en.wikipedia.org/wiki/Prime_k-tuple"]k-tuple[/URL], there is an infinite sequence of primes. It also implies that for k=2 and any even number, there is an infinite number of twin prime pairs, cousin primes, ect. However it is just a [URL="http://en.wikipedia.org/wiki/Conjecture"]conjecture[/URL]. A conjecture is something that a mathematician thinks to probably be true. For k=2 and a separation of 246, there is a theorem about pairs of primes. A summary about prime pairs is [URL="http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes"]here[/URL].

[URL="http://www.mersenneforum.org/showthread.php?t=16705&highlight=tuples"]Puzzle Peter and others[/URL] have been working on 3 and 4-tuples. k=3,4. But an enumeration of primes is not a theorem. It is just something else.

The Online Encyclopedia of Integer Sequences has a enumeration in [URL="http://oeis.org/A001359"]A1359[/URL]. 100,000 of the smallest twins are [URL="http://oeis.org/A001359/b001359.txt"]enumerated[/URL].

I have been working on prime constellations that is to say k-tuples for k up to 12. My webpage is [URL="https://sites.google.com/site/primeconstellations/"]here[/URL].

If I had to guess, I would say that the k-tuple conjecture is probably true. I wouldn't even know where to look for a counter-example.

Regards,
Matt

[QUOTE=lavalamp;399258]It seems that everyone here pretty much expects there to be an infinite number of twin primes. However, What do people think about the possible infinitude of cousin primes, sexy primes etc.? Might there be an infinite number of primes p and p+n where n is any even number?

Along similar lines perhaps, the Green-Tao theorem states that there are arbitrarily long arithmetic progressions of primes. I could pick "Graham's number" as the length and surely one of that length must exist, but does this extend to arithmetic progressions of infinite length?

Also, does this theorem or any other give an indication to the number of sequences of length k? Might there be an infinite number of each such sequence?[/QUOTE]
Terry Tao mentioned a new paper:
[url]https://terrytao.wordpress.com/2015/11/16/chains-of-large-gaps-between-primes/[/url]
[QUOTE]Kevin Ford, James Maynard, and I have uploaded to the arXiv our preprint “[URL="http://arxiv.org/abs/1511.04468"]Chains of large gaps between primes[/URL]“...[/QUOTE]
I'm not sure if this is relevant to any of the places on this forum that look at prime gaps but it might be.