Editing Cunningham chain (section)

==Definition==

A '''Cunningham chain of the first kind''' of length ''n'' is a sequence of prime numbers (''p''<sub>1</sub>, ..., ''p''<sub>''n''</sub>) such that ''p''<sub>''i''+1</sub>&nbsp;=&nbsp;2''p''<sub>''i''</sub>&nbsp;+&nbsp;1 for all 1&nbsp;≤&nbsp;''i''&nbsp;<&nbsp;''n''. (Hence each term of such a chain except the last is a [[Sophie Germain prime]], and each term except the first is a [[safe prime]]).

It follows that

: <math>
\begin{align}
p_2 & = 2p_1+1, \\
p_3 & = 4p_1+3, \\
p_4 & = 8p_1+7, \\
& {}\  \vdots \\
p_i & = 2^{i-1}p_1 + (2^{i-1}-1),
\end{align}
</math>

or, by setting <math>a = \frac{p_1 + 1}{2}</math> (the number <math>a</math> is not part of the sequence and need not be a prime number), we have <math>p_i = 2^{i} a - 1.</math>

Similarly, a '''Cunningham chain of the second kind''' of length ''n'' is a sequence of prime numbers (''p''<sub>1</sub>, ..., ''p''<sub>''n''</sub>) such that ''p''<sub>''i''+1</sub>&nbsp;=&nbsp;2''p''<sub>''i''</sub>&nbsp;−&nbsp;1 for all 1&nbsp;≤&nbsp;''i''&nbsp;<&nbsp;''n''.

It follows that the general term is

: <math>p_i = 2^{i-1}p_1 - (2^{i-1}-1).</math>

Now, by setting <math>a = \frac{p_1 - 1}{2} </math>, we have <math> p_i = 2^{i} a + 1</math>.

Cunningham chains are also sometimes generalized to sequences of prime numbers (''p''<sub>1</sub>, ..., ''p''<sub>''n''</sub>) such that ''p''<sub>''i''+1</sub> =&nbsp;''ap''<sub>''i''</sub>&nbsp;+&nbsp;''b'' for all 1&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''n'' for fixed [[coprime]] [[integer]]s ''a'' and ''b''; the resulting chains are called '''generalized Cunningham chains'''.

A Cunningham chain is called '''complete''' if it cannot be further extended, i.e., if the previous and the next terms in the chain are not prime numbers.