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Recall that a Sophie Germain prime is a prime p such that
q=2p+1 is also prime. Why not also ask that
r=2q+1 is prime, and 2r+1 is prime, and...?
A Cunningham chain of length k (of the first kind)
is sequence of k primes, each which is twice the preceding
one plus one. For example, {2, 5, 11, 23, 47} and {89, 179, 359, 719, 1439, 2879}.
A Cunningham chain of length k (of the
second kind) is a sequence of k primes, each which is
twice the preceding one minus one. (For example, {2, 3, 5} and
{1531, 3061, 6121, 12241, 24481}.)
Primes of both these forms
are called complete chains if they can not be extended
by adding either the next larger, or smaller, terms. In the following table (from [Loh89]) we list the first
complete chains for several lengths.
Prime at which the first complete chain starts
length  first kind  second kind 
1  13  11 
2  3  7 
3  41  2 
4  509  2131 
5  2  1531 
6  89  385591 
7  1122659  16651 
8  19099919  15514861 
9  85864769  857095381 
10 
26089808579  205528443121 
11 
665043081119  1389122693971 
12 
554688278429  216857744866621 
13 
4090932431513069  758083947856951 
How long can these chains get? The prime ktuple conjecture implies that there should be infinitely many for each of these primes. In fact, the number less than x should
be asymptotic to
where
The sequence B_{k} begins
approximately 1.32032 (k=2), 2.85825, 5.553491, 20.2636, 71.9622, 233.878, 677.356.
Tony Forbes has found chains of length 14 (for the first kind)
and 16 (for the second kind). See the links below for current records.
Note that some authors extend the definition of
Cunningham Chain to all sequences of primes
p_{i} the form
p_{i+1} =
ap_{i}+b
where a and b are fixed, relatively prime
integers with a > 1.
Related pages (outside of this work) References:
 Cunningham1907
 A. Cunnningham, "On hypereven numbers and on Fermat's numbers," Proc. Lond. Math. Soc., series 2, 5 (1907) 237274.
 Guy94 (SectionA7)
 R. K. Guy, Unsolved problems in number theory, SpringerVerlag, New York, NY, 1994. ISBN 0387942890. MR 96e:11002 [An excellent resource! Guy briefly describes many open questions, then provides numerous references. See his newer editions of this text.]
 Lehmer1965
 D. H. Lehmer, "On certain chains of primes," Proc. Lond. Math. Soc., series 3, 14a (1965) 183186. MR 31:2222
 Loh89
 G. Löh, "Long chains of nearly doubled primes," Math. Comp., 53 (1989) 751759. MR 90e:11015 (Abstract available) [Chains of primes for which each is either twice the proceeding one plus one, or each is either twice the proceeding one minus one. See also [Guy94, section A7].]
 Ribenboim95 (p 333)
 P. Ribenboim, The new book of prime number records, 3rd edition, SpringerVerlag, 1995. New York, NY, pp. xxiv+541, ISBN 0387944575. MR 96k:11112 [An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventyfive pages.]
 Yates82
 S. Yates, Repunits and repetends, Star Publishing Co., Inc., 1982. Boynton Beach, Florida, pp. vi+215, MR 83k:10014
