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. Eric W. Weisstein, ''CRC Concise Encyclopedia of Mathematics'', CRC Press, 2010, page 249.

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.

Relation with other properties

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.

Notes and references

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Prime numbers

{{Prime number classes