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 .