Editing Cunningham Project (section)

===Other factors===
Once the algebraic and aurifeuillean factors are removed, the other factors of {{math|''b''<sup>''n''</sup> ± 1}} are always of the form {{math|2''kn'' + 1}}, since the factors of {{math|''b''<sup>''n''</sup> − 1}} are all factors of <math>\Phi_n(b)</math>, and the factors of {{math|''b''<sup>''n''</sup> + 1}} are all factors of <math>\Phi_{2n}(b)</math>. When ''n'' is [[prime number|prime]], both algebraic and aurifeuillean factors are not possible, except the trivial factors ({{math|''b'' − 1}} for {{math|''b''<sup>''n''</sup> − 1}} and {{math|''b'' + 1}} for {{math|''b''<sup>''n''</sup> + 1}}). For [[Mersenne numbers]], the trivial factors are not possible for {{nowrap|prime ''n''}}, so all factors are of the form {{math|2''kn'' + 1}}. In general, all factors of {{math|(''b''<sup>''n''</sup> − 1)&hairsp;/(''b'' − 1)}} are of the form {{math|2''kn'' + 1,}} where {{math|''b'' ≥ 2}} and ''n'' is prime, except when ''n'' divides {{math|''b'' − 1}}, in which case {{math|(''b''<sup>''n''</sup> − 1)&hairsp;/(''b'' − 1)}} is divisible by ''n'' itself.

Cunningham numbers of the form {{math|''b''<sup>''n''</sup> − 1}} can only be prime if ''b'' = 2 and ''n'' is prime, assuming that ''n'' ≥ 2; these are the Mersenne numbers. Numbers of the form {{math|''b''<sup>''n''</sup> + 1}} can only be prime if ''b'' is even and ''n'' is a [[power of 2]], again assuming {{math|''n'' ≥ 2;}} these are the generalized Fermat numbers, which are [[Fermat number]]s when ''b'' = 2. Any factor of a Fermat number {{math|2<sup>2<sup>''n''</sup></sup> + 1}} is of the form {{math|''k''&middot;2<sup>''n''+2</sup> + 1}}.