Editing Cunningham chain (section)

==Examples==

Examples of complete Cunningham chains of the first kind include these:

: 2, 5,&thinsp;11, 23, 47 (The next number would be 95, but that is not prime.)
: 3, 7 (The next number would be 15, but that is not prime.)
: 29, 59 (The next number would be 119, but that is not prime.)
: 41, 83, 167 (The next number would be 335, but that is not prime.)
: 89, 179, 359, 719, 1439, 2879 (The next number would be 5759, but that is not prime.)

Examples of complete Cunningham chains of the second kind include these:

: 2, 3, 5 (The next number would be 9, but that is not prime.)
: 7, 13 (The next number would be 25, but that is not prime.)
: 19, 37, 73 (The next number would be 145, but that is not prime.)
: 31, 61 (The next number would be 121 = 11<sup>2</sup>, but that is not prime.)

Cunningham chains are now considered useful in cryptographic systems since "they provide two concurrent suitable settings for the [[ElGamal encryption|ElGamal cryptosystem]] ... [which] can be implemented in any field where the [[discrete logarithm problem]] is difficult."<ref>Joe Buhler, ''Algorithmic Number Theory: Third International Symposium, ANTS-III''. New York: Springer (1998): 290</ref>