In number theory, a bi-twin chain of length k + 1 is a sequence of natural numbers
n − 1 , n + 1 , 2 n − 1 , 2 n + 1 , … , 2 k n − 1 , 2 k n + 1 in which every number is prime.
The numbers n − 1 , 2 n − 1 , … , 2 k n − 1 form a Cunningham chain of the first kind of length k + 1 , while n + 1 , 2 n + 1 , … , 2 k n + 1 forms a Cunningham chain of the second kind. Each of the pairs 2 i n − 1 , 2 i n + 1 is a pair of twin primes. Each of the primes 2 i n − 1 for 0 ≤ i ≤ k − 1 is a Sophie Germain prime and each of the primes 2 i n − 1 for 1 ≤ i ≤ k is a safe prime.
q# denotes the primorial 2×3×5×7×...×q.
As of 2014, the longest known bi-twin chain is of length 8.
Cunningham chainTwin primesSophie Germain prime is a prime p such that 2 p + 1 is also prime.Safe prime is a prime p such that ( p − 1 ) / 2 is also prime.