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, ... (Sloane's 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 .
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. 
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, ... (Sloane's A003627). 
. 
for 
, and 
a prime congruent to 2 (mod 3). 
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 
.