Cunningham Chain

A Sequence of Primes $q_1 < q_2 < \ldots < q_k$ 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.

The two largest known Cunningham chains (of the first kind) of length three are ( $384205437\cdot 2^{4000}-1$, $384205437\cdot 2^{4001}-1$, $384205437\cdot 2^{4002}-1$) and ( $651358155\cdot 2^{3291}-1$, $651358155\cdot 2^{3292}-1$, $651358155\cdot 2^{3293}-1$), both discovered by W. Roonguthai in 1998.

See also Prime Arithmetic Progression, Prime Cluster

References

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.

Roonguthai, W. ``Yves Gallot's Proth and Cunningham Chains.'' http://ksc9.th.com/warut/cunningham.html.