Find Prime Double Pairs: Frequency & Infinity

[mathman]

Starting at 10, for any set of 5 consecutive odd numbers, at most 4 can be prime (the number ending in 5 cannot be prime). Moreover any such set has to have the number ending in 5 as the middle of two pairs of prime (you cannot have 3 consecutive odd primes when you start after 10). The first example of such a set is 11, 13, 17, 19. The next is 101, 103, 107, 109. How frequently does such a sequence occur? Is it known if there are an infinite number of such pairs of pairs?

 

[hello3719]

mostly related with the twin prime conjecture. If there exists an infinite of twin primes
( I am quite sure there is ) then IT IS possible that there exist an infinite of such sequences. Still an open problem, il try to close it this summer.

 

[Zurtex]

Recent work has shown there exists an infinite number of arithmetic series within primes.

Using your example of 11, 13, 17, 19. Is the same as 9 + 2n for n = 1, 2, 3, 4. Saying in this sequence that the number of terms is 4, t = 4, the proof shows that there exists in primes arithmetic series of the form a + dn for all t. t = 22 is the largest that has ever been calculated:

11,410,337,850,553 + 4,609,098,694,200n

for n = 1, 2, 3 … 22

However, the twin prime conjecture may be close to being solved, read here: http://mathworld.wolfram.com/news/2004-06-09/twinprimes/

Sorry but I have not heard of your problem before.

 

[shmoe]

Hi, 11, 13, 17, 19 isn't quite an arithmetic progression. B. Green & Tao's result, exciting though it is, on arbitrarily long arithmetic progressions of primes won't help here since it 'just' guarantees a progression of primes of the form n+dt for t=1..k for whatever value of k you like but you have no control over d. (it's actually slightly stronger- it guarantees such a progression in any subset of the primes that's dense enough)

Unfortunately I don't know much about mathman's problem apart from the obvious connections to the twin primes conjecture. I can't think of anything right off that would prevent infinitely many such sequences.

edit- http://mathworld.wolfram.com/PrimeConstellation.html gives the Hardy-Littlewood conjecture for the frequency of primes clusters you're looking at.