Problems & Puzzles: Puzzles

Puzzle 71.- Consecutive primes and Cunningham chains

A Cunningham chain of 1st order and length L is a sequence of L numbers: p, 2p + 1, 4p + 3, 8p + 7, 2^(L - 1)p + 2^(L - 1) - 1, such that all the numbers are prime.

A Cunningham chain of 2nd order and length L is a sequence of L numbers: p, 2p - 1, 4p - 3, 8p - 7, 2^(L - 1)p - (2^(L - 1) - 1), such that all the numbers are prime.

Below you can find my little search for the least sets of K consecutive primes for Cunningham chains of length 2 <= L <= 4, first and second order.

1st Order L=2 L=3 L=4
K  p such that p & 2p + 1 are primes p such that p, 2p + 1, &  4p+3 are primes p such that p, 2p + 1, 4p +3 & 8p +7 are primes
2 2, 3 1889, 1901 34139879, 34139909
3 2, 3, 5 66961409, 66961439, 66961451 ?
4 1433849,  1433891,  1433903,  1433909 ? ?
5 9816899,  9816923,  9816941,  9816953,  9816959 ? ?
6 445480319,  445480331,  445480361,  445480391,  445480421,  445480439  (This is my A047984) ? ?
7 ? ? ?

 

2nd Order L=2 L=3 L=4
K  p such that p & 2p - 1 are primes P such that p, 2p - 1, &  4p - 3 are primes P such that p, 2p - 1, 4p - 3 & 8p - 7 are primes
2 31, 37 25609   25621 66954961        66954991
3 3169,  3181,  3187 26923669        26923681        26923711 ?
4 63199,  63211,  63241,  63247 ? ?
5 17742877,  17742889,  17742919, 17742931,  17742937 ? ?
 

6

 

86257279,  86257387,  86257411,  86257417,  86257459,  86257489 ? ?
7 ? ? ?

Questions: Can you extend and/or complete the table?


Jud McCranie (17/10/99) sent the following comments: " I worked on puzzle 71, but didn't find anything new.  I tried L=2 - first order and second order, p<2^32.  I didn't find any longer sequences.  I verified your results for L=2 in both cases, except that for 2nd order 2 & 3 are the smallest solutions instead of 31 & 37"

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Giovanni Resta wrote (Nov. 2004):

(Consecutive primes and Cunningham chains) 1st order:
L=2 k=7 (298098924131, 298098924143, 298098924173, 298098924209,
298098924251, 298098924341, 298098924443)
L=3 k=4 (58308965339, 58308965369, 58308965381, 58308965411)
L=5 k=2 (5133129899, 5133129929)
(no other new solutions for L<=5 and p<=321,839,692,501)

2nd order:
L=5 k=2 (5412418021, 5412418051)
(no other new solutions for L<=5 and p<=305,882,832,161)
 

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