An Eisenstein prime is a nonzero Eisenstein integer that is not an Eisenstein unit and cannot be written
as a product of two Eisenstein
integers neither of which is an Eisenstein unit.
Writing 
for a root of unity of order 3, Eisenstein
integers have the form 
for integers 
and 
.
The Eisenstein primes 
with complex modulus 
are given by 
, 
, 
, 
, 
, 2, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, and 
, together with their negatives. 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).
Up to multiplication by an Eisenstein unit, the Eisenstein primes fall into three classes (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 
.
