
Glossary: Prime Pages: Top 5000: 
GIMPS has discovered a new largest known prime number: 2^{82589933}1 (24,862,048 digits) 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.
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.
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Chris K. Caldwell © 19992020 (all rights reserved)
