Bi-twin chain

In number theory, a bi-twin chain of length k + 1 is a sequence of natural numbers


 * $$ n-1,n+1,2n-1,2n+1, \dots, 2^k n - 1, 2^k n + 1 \,$$

in which every number is prime.

The special case, when the four numbers $$n-1,n+1,2n-1,2n+1$$ are all primes, they are called bi-twin primes, such n values are


 * 6, 30, 660, 810, 2130, 2550, 3330, 3390, 5850, 6270, 10530, 33180, 41610, 44130, 53550, 55440, 57330, 63840, 65100, 70380, 70980, 72270, 74100, 74760, 78780, 80670, 81930, 87540, 93240, …

Except 6, all of these numbers are divisible by 30.

The numbers $$n-1, 2n-1, \dots, 2^kn - 1$$ form a Cunningham chain of the first kind of length $$k + 1$$, while $$n+1, 2n + 1, \dots, 2^kn + 1$$ forms a Cunningham chain of the second kind. Each of the pairs $$2^in - 1, 2^in+ 1$$ is a pair of twin primes. Each of the primes $$2^in - 1$$ for $$0 \le i \le k - 1$$ is a Sophie Germain prime and each of the primes $$2^in - 1$$ for $$1 \le i \le k$$ is a safe prime.

Largest known bi-twin chains
q# denotes the primorial 2×3×5×7×...×q.

, the longest known bi-twin chain is of length 8.

Related chains

 * Cunningham chain

Related properties of primes/pairs of primes

 * Twin primes
 * Sophie Germain prime is a prime $$p$$ such that $$2p + 1$$ is also prime.
 * Safe prime is a prime $$p$$ such that $$(p-1)/2$$ is also prime.