A Cunningham chain, named after a mathematician by that name, is a sequence of prime numbers formed by successively doubling the initial prime and either adding 1 (Cunningham chain of the first type) or subtracting 1 (Cunningham chain of the second type) to form new primes. The chain terminates with the last prime for which the process produces a composite number.
For example: 2, 5, 11, 23, 47 form a chain of the first type with length 5 while 2, 3, 5 form a chain of the second type with length 3. All primes in a Cunningham chain are Sophie-Germain primes except the last and safe primes except the first. A Sophie-Germain prime is a prime \(p\) such that 2\(p\)+1 is also prime e.g. 5 is such a prime because 5 x 2+1=11. A safe prime is a prime \(p\) such that (\(p\)-1)/2 is also prime e.g. 23 is such a prime because (23-1)/2=11.
The cryptocurrency Primecoin whose logo appears in this post makes use of Cunningham chains of both types. Such a cryptocurrency has connections with Anarcho-capitalism, a term much beloved by Jeff Berwick and defined by Wikipedia as:
a political philosophy that advocates the elimination of political government - which distorts market signals, breeds corruption, and institutionalises monopoly - in favour of individual sovereignty, open markets (laissez-faire capitalism) and absence of invasive private property policies.
Additional posts regarding these sorts of chains can be found at:
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