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"
***
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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