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 chain
Twin primes
Sophie 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.
