Cunningham Chain

A sequence of primes  is a Cunningham chain of the first kind (second kind) of length  if  () for , ..., . Cunningham primes of the first kind are Sophie Germain primes.

It is conjectured there are arbitrarily long Cunningham chains. The longest known Cunningham chains are of length 17, with the first examples found corresponding to  (first kind; J. Wroblewski, May 2008) and  (second kind; J. Wroblewski, Jun. 2008).

The smallest prime beginning a complete Cunningham chain of the first kind of lengths , 2, ... are 13, 3, 41, 509, 2, 89, 1122659, 19099919, 85864769, 26089808579, ... (OEIS A005602).

The smallest prime beginning a complete Cunningham chain of the second kind of lengths , 2, ... are 11, 7, 2, 2131, 1531, 33301, 16651, 15514861, 857095381, 205528443121, ... (OEIS A005603).

REFERENCES:

Augustin, D. "Cunningham Chain Records." Dec. 30, 2008. https://hjem.get2net.dk/jka/math/Cunningham_Chain_records.htm.

Caldwell, C. "The Top Twenty: Cunningham Chain (1st Kind)." https://primes.utm.edu/top20/page.php?id=19.

Caldwell, C. "The Top Twenty: Cunningham Chain (2nd Kind)." https://primes.utm.edu/top20/page.php?id=20.

Forbes, T. "Prime Clusters and Cunningham Chains." Math. Comput. 68, 1739-1748, 1999.

Guy, R. K. "Cunningham Chains." §A7 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 18-19, 1994.

Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, p. 333, 1996.

Sloane, N. J. A. Sequences A005602 and A005603 in "The On-Line Encyclopedia of Integer Sequences."