An Eisenstein integer is a + bω, where a and b are both integers.

The imaginary unit for Eisenstein integers, where

ω == (-1/2) + ((sqrt 3)/2)ι == exp(2*pi*ι/3)

and ι is the usual imaginary unit with ι² == -1.

Conjugate a Eisenstein integer.

The square of the magnitude of a Eisenstein integer.

Produce a list of an EisensteinInteger's associates.

List of all Eisenstein units, counterclockwise across all sextants, starting with 1.

Find an Eisenstein integer whose norm is the given prime number in the form 3k + 1 using a modification of the Hermite-Serret algorithm.

The maintainer Andrew Lelechenko derived the following:

• Each prime of the form 3n + 1 is actually of the form 6k + 1.
• One has (z + 3k)^2 ≡ z^2 + 6kz + 9k^2 ≡ z^2 + (6k + 1)z - z + 9k^2 ≡ z^2 - z + 9k^2 (mod 6k + 1).

The goal is to solve z^2 - z + 1 ≡ 0 (mod 6k + 1). One has:

1. z^2 - z + 1 ≡ 0 (mod 6k + 1)

2. z^2 - z ≡ -1 (mod 6k + 1)

3. z^2 - z + 9k^2 ≡ 9k^2 - 1 (mod 6k + 1)

4. (z + 3k)^2 ≡ 9k^2 - 1 (mod 6k + 1)

5. z + 3k = sqrtsModPrime(9k^2 - 1) (mod 6k + 1)

6. z = (sqrtsModPrime(9k^2 - 1) (mod 6k + 1)) - 3k

For example, let p = 7, then k = 1. Square root of 9*1^2-1 ≡ 1 (mod 7), and z = 1 - 3*1 = -2 ≡ 5 (mod 7).

Truly, norm (5 :+ 1) = 25 - 5 + 1 = 21 ≡ 0 (mod 7).

An infinite list of Eisenstein primes. Uses primes in Z to exhaustively generate all Eisenstein primes in order of ascending norm.

• Every prime is in the first sextant, so the list contains no associates.
• Eisenstein primes from the whole complex plane can be generated by applying associates to each prime in this list.