Bi-twin chain

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

in which every number is prime.

The special case, when the four numbers 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 form a Cunningham chain of the first kind of length , while forms a Cunningham chain of the second kind. Each of the pairs is a pair of twin primes. Each of the primes for is a Sophie Germain prime and each of the primes for 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.

Relation with other properties

Related chains

• Cunningham chain

Related properties of primes/pairs of primes

• Twin primes
• Sophie Germain prime is a prime such that is also prime.
• Safe prime is a prime such that is also prime.

Notes and references