Prime double pairs.

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
( im 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/20...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.