Yes, there are such conjectures: see the work over the past few years by Terry Tao and the polymath group for lots of details.  Michael Nielsen maintains a site about this at http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes

What was the exact statement of those conjectures?

Thanks,

DG

No prime conjecture that I know of. Are you proposing one?

I am not sure that there is some concrete conjecture for the twin twins of primes or not. Of course, if the twin twins conjecture is true, then the twin primes conjecture is true. There are two twins next to each other.

If we require the distance between two primes in the multiplet are either 2 or 4, and the length is minimal,

then except the (3, 5, 7), the triplets could only be the upper twin triplets and the lower

twin triplets. The quadruplets could only be p, p+2, p+6, p+8 (in general). They could only be

a1, a3, a7, a9. Where a is literally an integer, e.g., a1 is actually 10a+1.

The quintuplets have only two possible forms

p, p+2, p+6, p+8, p+12 as one example I mentioned earlier with a form

a1, a3, a7, a9, (a+1)3. Or p-4, p, p+2, p+6, p+8. they could only be (a-1)7, a1, a3, a7, a9 (of course in general except the parts of the only eight-let).

There is only one general form of the six-let (a-1)7, a1, a3, a7, a9, (a+1)3.

Six is the highest possible except the single eight-let. So one might have a six-let conjecture also. Or, you only have finite many of them and therefore could be classified.

It would be interesting to give more examples of these multiplets.

So far, I only have one six-let as I mentioned above.

There's a nice article here from Wired which discusses related topics.  It uses the term "comb", and discusses the "prime k-tuple conjecture", namely that any "admissible comb" will hit primes infinitely often.

this http://www.wired.com/2013/11/prime/

Got it. See;

https://en.wikipedia.org/wiki/Prime_k-tuple#Named_patterns

Thanks,

DG

The Hardy-Littlewood Conjectures give a more refined version of the prime k-tuple conjecture. The k-tuples conjecture elicits an answer to the following: Given a k-tuple H = {h_1, ..., h_k} of distinct integers, are there infinitely many sets of the form n + H which contain only primes? We'll as you have noticed, it doesn't make sense to ask this question of all possible H. The 3-tuple H = {0, 2, 4} is a bad set on which to base this question. Why? Because the set n + H = {n, n + 2, n + 4} forms a complete residue system modulo 3. So, it must contain a number divisible by 3. We call a set H which avoids forming a complete residue system modulo a prime admissible. In practice, given a k-tuple H, one checks that H is admissible by verifying that H does not form a complete residue system modulo p for all primes p

Thank you for all the answers. My field is not number theory. So, I do not really understand the more abstract stuff. However, it is very interesting that when

H={1, 3, 7, 9} and {7, 11, 13, 17, 19, 23}.

Dear Patrick, what does the Hardy-Littlewood Conjecture say in these two cases?

Goldston, Daniel A.; Pintz, János; Yıldırım, Cem Y. Primes in tuples. I. Ann. of Math. (2) 170 (2009), no. 2, 819–862 proved in the second case, there are at least two primes for infinite many n under an assumption of another conjecture. Even though they are still short of proving all of them are primes.

I am wondering that if one just restrict his or her attention to these two cases, possibly he or she could get some useful results.

By the way, the proof for either of these two cases implies the twin prime conjecture. On the other hand, if someone could prove the second case is wrong, that would be also very interesting!

For the case H={0, 2}, the best result is due to Chen, Jingrun.

He proved that for any even h, {p, p+h} with p a prime hit  infinite many p+h

which is either a prime or an almost prime, i.e., the product of two primes.

Also, it seems to me that the Goldbach Conjecture is much harder than the twin

prime conjecture. Chen's result for that is seemly only:

There is an integer N such that for any even integer n>N there is a prime p