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In number theory, a bi-twin chain of length k + 1 is a sequence of natural numbers
in which every number is prime.
The numbers n − 1 , 2 n − 1 , … , 2 k n − 1 {\displaystyle n-1,2n-1,\dots ,2^{k}n-1} form a Cunningham chain of the first kind of length k + 1 {\displaystyle k+1} , while n + 1 , 2 n + 1 , … , 2 k n + 1 {\displaystyle n+1,2n+1,\dots ,2^{k}n+1} forms a Cunningham chain of the second kind. Each of the pairs 2 i n − 1 , 2 i n + 1 {\displaystyle 2^{i}n-1,2^{i}n+1} is a pair of twin primes. Each of the primes 2 i n − 1 {\displaystyle 2^{i}n-1} for 0 ≤ i ≤ k − 1 {\displaystyle 0\leq i\leq k-1} is a Sophie Germain prime and each of the primes 2 i n − 1 {\displaystyle 2^{i}n-1} for 1 ≤ i ≤ k {\displaystyle 1\leq i\leq k} is a safe prime.