Editing Mersenne prime (section)

===Complex numbers===
In the [[ring (mathematics)|ring]] of integers (on [[real number]]s), if {{math|''b'' − 1}} is a [[unit (ring theory)|unit]], then {{math|''b''}} is either 2 or 0. But {{math|2<sup>''n''</sup> − 1}} are the usual Mersenne primes, and the formula {{math|0<sup>''n''</sup> − 1}} does not lead to anything interesting (since it is always −1 for all {{math|''n'' > 0}}). Thus, we can regard a ring of "integers" on [[complex number]]s instead of [[real number]]s, like [[Gaussian integer]]s and [[Eisenstein integer]]s.

====Gaussian Mersenne primes====
If we regard the ring of [[Gaussian integer]]s, we get the case {{math|''b'' {{=}} 1 + ''i''}} and {{math|''b'' {{=}} 1 − ''i''}}, and can ask ([[without loss of generality|WLOG]]) for which {{math|''n''}} the number {{math|(1 + ''i'')<sup>''n''</sup> − 1}} is a [[Gaussian prime]] which will then be called a '''Gaussian Mersenne prime'''.<ref>Chris Caldwell: [http://primes.utm.edu/glossary/xpage/GaussianMersenne.html The Prime Glossary: Gaussian Mersenne] (part of the [[Prime Pages]])</ref>

{{math|(1 + ''i'')<sup>''n''</sup> − 1}} is a Gaussian prime for the following {{math|''n''}}:

:2, 3, 5, 7, 11, 19, 29, 47, 73, 79, 113, 151, 157, 163, 167, 239, 241, 283, 353, 367, 379, 457, 997, 1367, 3041, 10141, 14699, 27529, 49207, 77291, 85237, 106693, 160423, 203789, 364289, 991961, 1203793, 1667321, 3704053, 4792057, ... {{OEIS|id=A057429}}

Like the sequence of exponents for usual Mersenne primes, this sequence contains only (rational) prime numbers.

As for all Gaussian primes, the [[Gaussian integer#Norm of a Gaussian integer|norm]]s (that is, squares of absolute values) of these numbers are rational primes:

:5, 13, 41, 113, 2113, 525313, 536903681, 140737471578113, ... {{OEIS|id=A182300}}.

====Eisenstein Mersenne primes====
One may encounter cases where such a Mersenne prime is also an ''Eisenstein prime'', being of the form {{math|''b'' {{=}} 1 + ''ω''}} and {{math|''b'' {{=}} 1 − ''ω''}}. In these cases, such numbers are called '''Eisenstein Mersenne primes'''.

{{math|(1 + ''ω'')<sup>''n''</sup> − 1}} is an Eisenstein prime for the following {{math|''n''}}:

:2, 5, 7, 11, 17, 19, 79, 163, 193, 239, 317, 353, 659, 709, 1049, 1103, 1759, 2029, 5153, 7541, 9049, 10453, 23743, 255361, 534827, 2237561, ... {{OEIS|id=A066408}}

The norms (that is, squares of absolute values) of these Eisenstein primes are rational primes:

:7, 271, 2269, 176419, 129159847, 1162320517, ... {{OEIS|id=A066413}}