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Cunningham Chain


A sequence of primes q_1<q_2<...<q_k is a Cunningham chain of the first kind (second kind) of length k if q_(i+1)=2q_i+1 (q_(i+1)=2q_i-1) for i=1, ..., k-1. 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 q_1=2759832934171386593519 (first kind; J. Wroblewski, May 2008) and q_1=40244844789379926979141 (second kind; J. Wroblewski, Jun. 2008).

The smallest prime beginning a complete Cunningham chain of the first kind of lengths n=1, 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 n=1, 2, ... are 11, 7, 2, 2131, 1531, 33301, 16651, 15514861, 857095381, 205528443121, ... (OEIS A005603).


See also

Bitwin Chain, Prime Arithmetic Progression, Prime Constellation

Portions of this entry contributed by Jens Kruse Andersen

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References

Augustin, D. "Cunningham Chain Records." Dec. 30, 2008. http://hjem.get2net.dk/jka/math/Cunningham_Chain_records.htm.Caldwell, C. "The Top Twenty: Cunningham Chain (1st Kind)." http://primes.utm.edu/top20/page.php?id=19.Caldwell, C. "The Top Twenty: Cunningham Chain (2nd Kind)." http://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."

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Cunningham Chain

Cite this as:

Andersen, Jens Kruse and Weisstein, Eric W. "Cunningham Chain." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CunninghamChain.html

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