Let 
be the cube root of unity 
. Then the Eisenstein primes are Eisenstein
integers, i.e., numbers of the form 
for 
and 
integers, such that 
cannot be written as a product of other Eisenstein
integers.
The Eisenstein primes 
with complex modulus 
are given by 
, 
, 
, 
, 
, 2, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, and 
. The positive Eisenstein primes with zero imaginary
part are precisely the ordinary primes that are congruent to 2 (mod 3), i.e., 2,
5, 11, 17, 23, 29, 41, 47, 53, 59, ... (OEIS A003627).
In particular, there are three classes of Eisenstein primes (Cox 1989; Wagon 1991, p. 320):
1. 
.
2. Numbers of the form 
for 
, and 
a prime congruent to 2 (mod
3).
3. Numbers of the form 
or 
where 
is a prime 
congruent to 1 (mod 3). Since primes of this form always have
the form 
,
finding the corresponding 
and 
gives 
and 
via 
and 
.
