A Cunningham chain of the first kind (second kind) of length is a sequence of primes satisfying () for , ..., . Each term except the last in a Cunningham chain of the
first kind is a Sophie Germain prime. A chain
is complete when no prime number can be prepended
or appended while preserving its defining recurrence
relation.
It is conjectured there are arbitrarily long Cunningham chains. As of Dec. 2008, the longest known Cunningham chains were of length 17 (Augustin 2008), with the first
examples found corresponding to (first kind; J. Wroblewski,
May 2008) and
(second kind; J. Wroblewski, Jun. 2008).
As of Oct. 2026, chains of length 20 of both kinds are recorded with for the first kind and for the second kind. Both are
attributed to J. Armitage ("Cunningham Chain Records" 2026).
The smallest primes beginning complete Cunningham chains of the first kind of lengths , 2, ... are 13, 3, 41, 509, 2, 89, 1122659, 19099919, 85864769,
26089808579, ... (OEIS A005602).
The smallest primes beginning complete Cunningham chains of the second kind of lengths , 2, ... are 11, 7, 2, 2131, 1531, 385591, 16651, 15514861,
857095381, 205528443121, ... (OEIS A005603).
is a sequence of primes 
satisfying 
(
) for 
, ..., 
. Each term except the last in a Cunningham chain of the
first kind is a Sophie Germain prime. A chain
is complete when no prime number can be prepended
or appended while preserving its defining recurrence
relation.
(first kind; J. Wroblewski,
May 2008) and 
(second kind; J. Wroblewski, Jun. 2008).
for the first kind and 
for the second kind. Both are
attributed to J. Armitage ("Cunningham Chain Records" 2026).
, 2, ... are 13, 3, 41, 509, 2, 89, 1122659, 19099919, 85864769,
26089808579, ... (OEIS A005602).
, 2, ... are 11, 7, 2, 2131, 1531, 385591, 16651, 15514861,
857095381, 205528443121, ... (OEIS A005603).
