Eisenstein primes: Difference between revisions

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Latest revision as of 18:36, 17 March 2024

An Eisenstein integer is a non-unit Gaussian integer a + bω where ω(1+ω) = -1, and a and b are integers.

ω is generally chosen as a cube root of unity: ${\displaystyle \omega ={\frac {-1+i{\sqrt {3}}}{2}}=e^{i2\pi /3}}$

As with a Gaussian integer, any Eisenstein integer is either a unit (an integer with a multiplicative inverse [±1, ±ω, ±(ω^-1)]), a prime (a number p such that if p divides xy, then p necessarily divides either x or y), or composite (a product of primes).

An Eisenstein integer a + bω is a prime if either it is a product of a unit and an integer prime p such that p % 3 == 2 or norm(a + bω) is an integer prime.

Eisenstein numbers can be generated by choosing any a and b such that a and b are integers. To allow generation in a relatively fixed order, such numbers can be ordered by their 2-norm or norm:

    norm(eisenstein integer(a, b)) = |a + bω|² =  $a^{2}-ab+b^{2}$

Task

Find, and show here as their complex number values, the first 100 (by norm order) Eisenstein primes nearest 0.

Stretch

Plot in the complex plane at least the first 2000 such numbers (again, as found with norm closest to 0).

See also

[Wikipedia entry for Eisenstein_integer] [Observable entry on Eisenstein primes]

J

Implementation:

eisensteinprimes=: {{
  rY=. >.1.5%:y
  p1=. ,(w^i.3)*/(#~ 2= 3|]) p:i.rY
  'a b'=. |:(2#rY)#:I.,1 p: {{(x*x)+(y*y)-x*y}}"0/~i.rY
  y{.(/: *:@|)p1,(,-)(a+b*w),a+b*-w
}}

Task example (and stretch - taking the stretch goal in a minimalist literal fashion):

   20 5$eisensteinprimes 100
     0j1.73205   1.5j0.866025     0j_1.73205 _1.5j_0.866025   _1j_1.73205
             2     _1j1.73205   _0.5j2.59808    0.5j2.59808  2.5j0.866025
     2j1.73205     2j_1.73205  2.5j_0.866025   0.5j_2.59808 _0.5j_2.59808
_2.5j_0.866025    _2j_1.73205     _2j1.73205  _2.5j0.866025     _1j3.4641
      1j3.4641      1j_3.4641     _1j_3.4641   3.5j0.866025   2.5j2.59808
  2.5j_2.59808  3.5j_0.866025 _3.5j_0.866025  _2.5j_2.59808  _2.5j2.59808
 _3.5j0.866025   _0.5j4.33013    0.5j4.33013    3.5j2.59808  3.5j_2.59808
  0.5j_4.33013  _0.5j_4.33013  _3.5j_2.59808   _3.5j2.59808     4j1.73205
    4j_1.73205    _4j_1.73205     _4j1.73205      3j_3.4641     _3j3.4641
 4.5j_0.866025  _4.5j0.866025  _2.5j_4.33013              5  _2.5j4.33013
    _2j5.19615      2j5.19615   5.5j0.866025    3.5j4.33013    2j_5.19615
   _2j_5.19615 _5.5j_0.866025  _3.5j_4.33013   _0.5j6.06218   0.5j6.06218
      5j3.4641      5j_3.4641   0.5j_6.06218  _0.5j_6.06218    _5j_3.4641
     _5j3.4641    5.5j2.59808   5.5j_2.59808  _5.5j_2.59808  _5.5j2.59808
  4.5j_4.33013   _4.5j4.33013     6j_1.73205     _6j1.73205  _2.5j6.06218
   2.5j6.06218   6.5j0.866025      4j5.19615     4j_5.19615 6.5j_0.866025
  2.5j_6.06218  _2.5j_6.06218 _6.5j_0.866025    _4j_5.19615    _4j5.19615
 _6.5j0.866025   5.5j_4.33013   6.5j_2.59808   _5.5j4.33013  _6.5j2.59808
  4.5j_6.06218  7.5j_0.866025   _4.5j6.06218  _7.5j0.866025  _0.5j7.79423
   0.5j7.79423    6.5j4.33013   0.5j_7.79423  _0.5j_7.79423 _6.5j_4.33013

   require'plot'
   'marker; markersize 0.3' plot eisensteinprimes 2000

jq

Works with jq, the C implementation of jq

Works with gojq, the Go implementation of jq

Adapted from Wren

In this entry, complex numbers are represented as arrays of pairs: [real, complex], as in the jq section on the Complex page. The two functions for adding and multiplying complex numbers presented there are reproduced below so that the program presented here is self-contained.

For the "stretch" task, we assume the availability of a tool such as gnuplot; using gnuplot, a suitable sequence of commands to plot the points produced by `graph` as defined below would be as follows:

reset
set terminal pngcairo
set output "eisenstein-primes.png"

### Complex numbers
def plus(x; y):
    if (x|type) == "number" then
       if  (y|type) == "number" then [ x+y, 0 ]
       else [ x + y[0], y[1]]
       end
    elif (y|type) == "number" then plus(y;x)
    else [ x[0] + y[0], x[1] + y[1] ]
    end;
 
def multiply(x; y):
    if (x|type) == "number" then
       if  (y|type) == "number" then [ x*y, 0 ]
       else [x * y[0], x * y[1]]
       end
    elif (y|type) == "number" then multiply(y;x)
    else [ x[0] * y[0] - x[1] * y[1],  x[0] * y[1] + x[1] * y[0]]
    end;

### Generic utilities

def lpad($len): tostring | ($len - length) as $l | (" " * $l) + .;

# Require $n > 0
def nwise($n):
  def _n: if length <= $n then . else .[:$n] , (.[$n:] | _n) end;
  if $n <= 0 then "nwise: argument should be non-negative" else _n end;

def is_prime:
  . as $n
  | if ($n < 2)         then false
    elif ($n % 2 == 0)  then $n == 2
    elif ($n % 3 == 0)  then $n == 3
    elif ($n % 5 == 0)  then $n == 5
    elif ($n % 7 == 0)  then $n == 7
    elif ($n % 11 == 0) then $n == 11
    elif ($n % 13 == 0) then $n == 13
    elif ($n % 17 == 0) then $n == 17
    elif ($n % 19 == 0) then $n == 19
    else sqrt as $s
    | 23
    | until( . > $s or ($n % . == 0); . + 2)
    | . > $s
    end;

def OMEGA: [-0.5, (3|sqrt * 0.5)];

### Eisenstein numbers and Eisenstein primes

def Eisenstein($a; $b):
   {$a, $b, n: plus( multiply(OMEGA;$b); $a) };

def realEisenstein: .n[0];
def imagEisenstein: .n[1];
def normEisenstein:
   .a *.a - .a * .b + .b * .b ;

# Replicate the Julia sort order for easy comparison
def sortEisenstein:
   sort_by( [ normEisenstein, imagEisenstein, realEisenstein] );

def isPrimeEisenstein:
    if .a == 0 or .b == 0 or .a == .b
    # length ~ abs
    then ([.a, .b] | map(length) | max) as $c  
    | ($c | is_prime) and $c % 3 == 2
    else normEisenstein | is_prime
    end;

# Eisenstein($i;$j) primes for $i and $j in -$n .. $n inclusive
def eprimes($n):
  reduce range (-$n; $n+1) as $a ([];
    reduce range ( -$n; $n+1) as $b (.;
      Eisenstein($a; $b) as $e
      | if $e | isPrimeEisenstein
        then . + [$e]
        else .
        end ));

### The tasks

# pretty-print a complex number
def pp:
  def r: 100 * . | trunc / 100;
  .[2] = (if .[1] < 0 then "-" else "+" end)
  | .[1] |= (if . < 0 then -. else . end)
  | "\(.[0]|r|lpad(5)) \(.[2]) \(.[1]|r|lpad(5))i";

# Display the input array of complex numbers as a table with $n columns
# proceeding row-wise and using pp/0
def row_wise($n):
  nwise($n) | map( pp ) | join("  ");

def listing:
  {eprimes: (eprimes(10) | sortEisenstein) }
  # convert to Complex numbers for easy display
  | .eprimes |= map( .n )
  | "First 100 Eisenstein primes nearest zero:",
    (.eprimes[:100] | row_wise(4) );

def graph:
  eprimes(100)
  | sortEisenstein
  | .[:2000][]
  | .n
  | "\(real(.)) \(imag(.))";

# For a listing of the first 100 Eisenstein primes nearest 0:
listing

# To produce the points for gnuplot:
# graph

Output:

The results of `listing` are shown below. For the graph of the output produced by `graph`, see the graph shown above at J.

First 100 Eisenstein primes nearest zero:
    0 -  1.73i   -1.5 -  0.86i    1.5 -  0.86i   -1.5 +  0.86i
  1.5 +  0.86i      0 +  1.73i     -1 -  1.73i      1 -  1.73i
   -2 +     0i      2 +     0i     -1 +  1.73i      1 +  1.73i
 -0.5 -  2.59i    0.5 -  2.59i     -2 -  1.73i      2 -  1.73i
 -2.5 -  0.86i    2.5 -  0.86i   -2.5 +  0.86i    2.5 +  0.86i
   -2 +  1.73i      2 +  1.73i   -0.5 +  2.59i    0.5 +  2.59i
   -1 -  3.46i      1 -  3.46i   -2.5 -  2.59i    2.5 -  2.59i
 -3.5 -  0.86i    3.5 -  0.86i   -3.5 +  0.86i    3.5 +  0.86i
 -2.5 +  2.59i    2.5 +  2.59i     -1 +  3.46i      1 +  3.46i
 -0.5 -  4.33i    0.5 -  4.33i   -3.5 -  2.59i    3.5 -  2.59i
   -4 -  1.73i      4 -  1.73i     -4 +  1.73i      4 +  1.73i
 -3.5 +  2.59i    3.5 +  2.59i   -0.5 +  4.33i    0.5 +  4.33i
 -2.5 -  4.33i    2.5 -  4.33i     -5 +     0i      5 +     0i
 -2.5 +  4.33i    2.5 +  4.33i     -2 -  5.19i      2 -  5.19i
 -3.5 -  4.33i    3.5 -  4.33i   -5.5 -  0.86i    5.5 -  0.86i
 -5.5 +  0.86i    5.5 +  0.86i   -3.5 +  4.33i    3.5 +  4.33i
   -2 +  5.19i      2 +  5.19i   -0.5 -  6.06i    0.5 -  6.06i
   -5 -  3.46i      5 -  3.46i   -5.5 -  2.59i    5.5 -  2.59i
 -5.5 +  2.59i    5.5 +  2.59i     -5 +  3.46i      5 +  3.46i
 -0.5 +  6.06i    0.5 +  6.06i   -2.5 -  6.06i    2.5 -  6.06i
   -4 -  5.19i      4 -  5.19i   -6.5 -  0.86i    6.5 -  0.86i
 -6.5 +  0.86i    6.5 +  0.86i     -4 +  5.19i      4 +  5.19i
 -2.5 +  6.06i    2.5 +  6.06i   -0.5 -  7.79i    0.5 -  7.79i
 -6.5 -  4.33i    6.5 -  4.33i     -7 -  3.46i      7 -  3.46i
   -7 +  3.46i      7 +  3.46i   -6.5 +  4.33i    6.5 +  4.33i

Julia

""" rosettacode.org/wiki/Eisenstein_primes """

import Base: Complex, real, imag
import LinearAlgebra: norm
import Primes: isprime
import Plots: scatter

struct Eisenstein{T<:Integer} <: Number
    a::T
    b::T
    Eisenstein(a::T, b::T) where {T} = new{T}(a, b)
    Eisenstein(a::T) where {T<:Integer} = new{T}(a, zero(T))
    Eisenstein(a::Integer, b::Integer) = new{eltype(promote(a, b))}(promote(a, b)...)
end

const ω = Eisenstein(false, true)

real(n::Eisenstein) = n.a - n.b / 2

imag(n::Eisenstein) = n.b * sqrt(big(3.0)) / 2

norm(n::Eisenstein) = n.a * n.a + n.b * n.b - n.a * n.b

Complex(n::Eisenstein{T}) where {T} = Complex{typeof(real(n))}(real(n), imag(n))

"""
    is_eisenstein_prime(n)

An Eisenstein integer is a non-unit Gaussian integer a + bω where ω(1+ω) = -1,
and a and b are integers. As a Gaussian integer, any Eisenstein integer is
either a unit (an integer with a multiplicative inverse [±1, ±ω, ±(ω^-1)]),
prime (a number p such that if p divides xy, then p necessarily divides
either x or y), or composite (a product of primes).

    An Eisenstein integer a + bω is a prime if either it is a product of a unit
and an integer prime p such that p % 3 == 2 or norm(a + bω) is an integer prime.
"""
function is_eisenstein_prime(n::Eisenstein)
    if n.a == 0 || n.b == 0 || n.a == n.b
        c = max(abs(n.a), abs(n.b))
        return isprime(c) && c % 3 == 2
    else
        return isprime(norm(n))
    end
end

function test_eisenstein_primes(graphlimitsquared = 10_000, printlimit = 100)
    lim = isqrt(graphlimitsquared)
    arr = [Eisenstein(a, b) for a = -lim:lim, b = -lim:lim]
    eprimes = sort!(filter(is_eisenstein_prime, arr), lt = (x, y) -> norm(x) < norm(y)))
    for (i, c) in enumerate(eprimes)
        if i <= printlimit
            print(lpad(round(Complex(c), digits = 4), 18), i % 5 == 0 ? "\n" : "")
        end
    end
    display(
        scatter(
            map(real, eprimes),
            map(imag, eprimes),
            markersize = 1,
            title = "Eisenstein primes with norm < $lim",
        ),
    )
end

test_eisenstein_primes()

Output:

    0.0 - 1.7321im    -1.5 - 0.866im     1.5 - 0.866im    -1.5 + 0.866im     1.5 + 0.866im
    0.0 + 1.7321im   -1.0 - 1.7321im    1.0 - 1.7321im      -2.0 + 0.0im       2.0 + 0.0im
   -1.0 + 1.7321im    1.0 + 1.7321im   -0.5 - 2.5981im    0.5 - 2.5981im   -2.0 - 1.7321im
    2.0 - 1.7321im    -2.5 - 0.866im     2.5 - 0.866im    -2.5 + 0.866im     2.5 + 0.866im
   -2.0 + 1.7321im    2.0 + 1.7321im   -0.5 + 2.5981im    0.5 + 2.5981im   -1.0 - 3.4641im
    1.0 - 3.4641im   -2.5 - 2.5981im    2.5 - 2.5981im    -3.5 - 0.866im     3.5 - 0.866im
    -3.5 + 0.866im     3.5 + 0.866im   -2.5 + 2.5981im    2.5 + 2.5981im   -1.0 + 3.4641im
    1.0 + 3.4641im   -0.5 - 4.3301im    0.5 - 4.3301im   -3.5 - 2.5981im    3.5 - 2.5981im
   -4.0 - 1.7321im    4.0 - 1.7321im   -4.0 + 1.7321im    4.0 + 1.7321im   -3.5 + 2.5981im
    3.5 + 2.5981im   -0.5 + 4.3301im    0.5 + 4.3301im   -2.5 - 4.3301im    2.5 - 4.3301im
      -5.0 + 0.0im       5.0 + 0.0im   -2.5 + 4.3301im    2.5 + 4.3301im   -2.0 - 5.1962im
    2.0 - 5.1962im   -3.5 - 4.3301im    3.5 - 4.3301im    -5.5 - 0.866im     5.5 - 0.866im
    -5.5 + 0.866im     5.5 + 0.866im   -3.5 + 4.3301im    3.5 + 4.3301im   -2.0 + 5.1962im
    2.0 + 5.1962im   -0.5 - 6.0622im    0.5 - 6.0622im   -5.0 - 3.4641im    5.0 - 3.4641im
   -5.5 - 2.5981im    5.5 - 2.5981im   -5.5 + 2.5981im    5.5 + 2.5981im   -5.0 + 3.4641im
    5.0 + 3.4641im   -0.5 + 6.0622im    0.5 + 6.0622im   -2.5 - 6.0622im    2.5 - 6.0622im
   -4.0 - 5.1962im    4.0 - 5.1962im    -6.5 - 0.866im     6.5 - 0.866im    -6.5 + 0.866im
     6.5 + 0.866im   -4.0 + 5.1962im    4.0 + 5.1962im   -2.5 + 6.0622im    2.5 + 6.0622im
   -0.5 - 7.7942im    0.5 - 7.7942im   -6.5 - 4.3301im    6.5 - 4.3301im   -7.0 - 3.4641im
    7.0 - 3.4641im   -7.0 + 3.4641im    7.0 + 3.4641im   -6.5 + 4.3301im    6.5 + 4.3301im

Nim

Translation of: Wren

Library: gnuplotlib.nim

import std/[algorithm, complex, math, strformat]

import gnuplot

func isPrime(n: Natural): bool =
  if n < 2: return false
  if (n and 1) == 0: return n == 2
  if n mod 3 == 0: return n == 3
  var k = 5
  var delta = 2
  while k * k <= n:
    if n mod k == 0: return false
    inc k, delta
    delta = 6 - delta
  result = true


### Eisenstein definition.

const ω = complex(-0.5, sqrt(3.0) * 0.5)

type Eisenstein = object
  a: int
  b: int
  n: Complex64

func initEisenstein(a, b: int): Eisenstein =
  ## Initialize an Eisenstein number.
  Eisenstein(a: a, b: b, n: a.toFloat + b.toFloat * ω)

template re(e: Eisenstein): float = e.n.re
template im(e: Eisenstein): float = e.n.im

func norm(e: Eisenstein): int =
  ## return the norm of an Eisenstein number.
  e.a * e.a - e.a * e.b + e.b * e.b

func isPrime(e: Eisenstein): bool =
  ## Return true if an Eisenstein number is prime.
  if e.a == 0 or e.b == 0 or e.a == e.b:
    let c = max(abs(e.a), abs(e.b))
    result = c.isPrime and c mod 3 == 2
  else:
    result = e.norm.isPrime

func `$`(e: Eisenstein): string =
  ## Return a string representation of an Eisenstein number.
  let (sign, im) = if e.im >= 0: ('+', e.im) else: ('-', -e.im)
  result =  &"{e.re:7.4f} {sign} {im:6.4f}i"


### Find Eisenstein primes.

var eprimes: seq[Eisenstein]
for a in -100..100:
  for b in -100..100:
    let e = initEisenstein(a, b)
    if e.isPrime: eprimes.add e

# Try to replicate Wren sort order for easy comparison.
eprimes.sort(proc (e1, e2: Eisenstein): int =
               result = cmp(e1.norm, e2.norm)
               if result == 0:
                 result = cmp(e1.im, e2.im)
                 if result == 0:
                   result = cmp(e1.re, e2.re)
            )

# Display first 100 Eisenstein primes to terminal.
echo "First 100 Eisenstein primes nearest zero:"
for i in 0..99:
  stdout.write eprimes[i]
  stdout.write if i mod 4 == 3: "\n" else: "  "

# Generate points for the plot.
var x, y: seq[float]
for e in eprimes:
  x.add e.re
  y.add e.im

withGnuPlot:
  cmd "set size ratio -1"
  plot(x, y, "Eisenstein primes", "with dots lw 2")

Output:

First 100 Eisenstein primes nearest zero:
 0.0000 - 1.7321i  -1.5000 - 0.8660i   1.5000 - 0.8660i  -1.5000 + 0.8660i
 1.5000 + 0.8660i   0.0000 + 1.7321i  -1.0000 - 1.7321i   1.0000 - 1.7321i
-2.0000 + 0.0000i   2.0000 + 0.0000i  -1.0000 + 1.7321i   1.0000 + 1.7321i
-0.5000 - 2.5981i   0.5000 - 2.5981i  -2.0000 - 1.7321i   2.0000 - 1.7321i
-2.5000 - 0.8660i   2.5000 - 0.8660i  -2.5000 + 0.8660i   2.5000 + 0.8660i
-2.0000 + 1.7321i   2.0000 + 1.7321i  -0.5000 + 2.5981i   0.5000 + 2.5981i
-1.0000 - 3.4641i   1.0000 - 3.4641i  -2.5000 - 2.5981i   2.5000 - 2.5981i
-3.5000 - 0.8660i   3.5000 - 0.8660i  -3.5000 + 0.8660i   3.5000 + 0.8660i
-2.5000 + 2.5981i   2.5000 + 2.5981i  -1.0000 + 3.4641i   1.0000 + 3.4641i
-0.5000 - 4.3301i   0.5000 - 4.3301i  -3.5000 - 2.5981i   3.5000 - 2.5981i
-4.0000 - 1.7321i   4.0000 - 1.7321i  -4.0000 + 1.7321i   4.0000 + 1.7321i
-3.5000 + 2.5981i   3.5000 + 2.5981i  -0.5000 + 4.3301i   0.5000 + 4.3301i
-2.5000 - 4.3301i   2.5000 - 4.3301i  -5.0000 + 0.0000i   5.0000 + 0.0000i
-2.5000 + 4.3301i   2.5000 + 4.3301i  -2.0000 - 5.1962i   2.0000 - 5.1962i
-3.5000 - 4.3301i   3.5000 - 4.3301i  -5.5000 - 0.8660i   5.5000 - 0.8660i
-5.5000 + 0.8660i   5.5000 + 0.8660i  -3.5000 + 4.3301i   3.5000 + 4.3301i
-2.0000 + 5.1962i   2.0000 + 5.1962i  -0.5000 - 6.0622i   0.5000 - 6.0622i
-5.0000 - 3.4641i   5.0000 - 3.4641i  -5.5000 - 2.5981i   5.5000 - 2.5981i
-5.5000 + 2.5981i   5.5000 + 2.5981i  -5.0000 + 3.4641i   5.0000 + 3.4641i
-0.5000 + 6.0622i   0.5000 + 6.0622i  -2.5000 - 6.0622i   2.5000 - 6.0622i
-4.0000 - 5.1962i   4.0000 - 5.1962i  -6.5000 - 0.8660i   6.5000 - 0.8660i
-6.5000 + 0.8660i   6.5000 + 0.8660i  -4.0000 + 5.1962i   4.0000 + 5.1962i
-2.5000 + 6.0622i   2.5000 + 6.0622i  -0.5000 - 7.7942i   0.5000 - 7.7942i
-6.5000 - 4.3301i   6.5000 - 4.3301i  -7.0000 - 3.4641i   7.0000 - 3.4641i
-7.0000 + 3.4641i   7.0000 + 3.4641i  -6.5000 + 4.3301i   6.5000 + 4.3301i

Perl

Translation of: Raku

use v5.36;
use Math::AnyNum <pi mod max complex reals is_prime>;

my $omega = exp ( complex(0,2) * pi/3 ); my @E;

sub    norm (@p)     { $p[0]**2 - $p[0]*$p[1] + $p[1]**2 }
sub display (@p)     { sprintf '%+8.4f%+8.4fi', reals($p[0] + $omega*$p[1]) }
sub       X ($a, $b) { my @p; for my $x ($a..$b) { for my $y ($a..$b) { push @p, [$x, $y] } } @p }
sub   table ($c, @V) { my $t = $c * (my $w = 1 + max map { length } @V); ( sprintf( ('%'.$w.'s')x@V, @V) ) =~ s/.{1,$t}\K/\n/gr }

for (X -10, 10) {
    my($a,$b) = @$_;
    my $c = max abs($a), abs($b);
    push @E, [@$_] if ((0==$a or 0==$b or $a==$b) and is_prime $c and 2 == mod $c,3) or is_prime norm @$_
}
say table 4, (map { display @$_ } sort { norm(@$a) <=> norm(@$b) } @E)[0..99];

Output:

  -1.5000 -0.8660i  -0.0000 -1.7321i  -1.5000 +0.8660i  +1.5000 -0.8660i
  +0.0000 +1.7321i  +1.5000 +0.8660i  -1.0000 -1.7321i  -2.0000 +0.0000i
  +1.0000 -1.7321i  -1.0000 +1.7321i  +2.0000 +0.0000i  +1.0000 +1.7321i
  -2.0000 -1.7321i  -2.5000 -0.8660i  -0.5000 -2.5981i  -2.5000 +0.8660i
  +0.5000 -2.5981i  -2.0000 +1.7321i  +2.0000 -1.7321i  -0.5000 +2.5981i
  +2.5000 -0.8660i  +0.5000 +2.5981i  +2.5000 +0.8660i  +2.0000 +1.7321i
  -2.5000 -2.5981i  -3.5000 -0.8660i  -1.0000 -3.4641i  -3.5000 +0.8660i
  +1.0000 -3.4641i  -2.5000 +2.5981i  +2.5000 -2.5981i  -1.0000 +3.4641i
  +3.5000 -0.8660i  +1.0000 +3.4641i  +3.5000 +0.8660i  +2.5000 +2.5981i
  -3.5000 -2.5981i  -4.0000 -1.7321i  -0.5000 -4.3301i  -4.0000 +1.7321i
  +0.5000 -4.3301i  -3.5000 +2.5981i  +3.5000 -2.5981i  -0.5000 +4.3301i
  +4.0000 -1.7321i  +0.5000 +4.3301i  +4.0000 +1.7321i  +3.5000 +2.5981i
  -2.5000 -4.3301i  -5.0000 +0.0000i  +2.5000 -4.3301i  -2.5000 +4.3301i
  +5.0000 +0.0000i  +2.5000 +4.3301i  -3.5000 -4.3301i  -5.5000 -0.8660i
  -2.0000 -5.1962i  -5.5000 +0.8660i  +2.0000 -5.1962i  -3.5000 +4.3301i
  +3.5000 -4.3301i  -2.0000 +5.1962i  +5.5000 -0.8660i  +2.0000 +5.1962i
  +5.5000 +0.8660i  +3.5000 +4.3301i  -5.0000 -3.4641i  -5.5000 -2.5981i
  -0.5000 -6.0622i  -5.5000 +2.5981i  +0.5000 -6.0622i  -5.0000 +3.4641i
  +5.0000 -3.4641i  -0.5000 +6.0622i  +5.5000 -2.5981i  +0.5000 +6.0622i
  +5.5000 +2.5981i  +5.0000 +3.4641i  -4.0000 -5.1962i  -6.5000 -0.8660i
  -2.5000 -6.0622i  -6.5000 +0.8660i  +2.5000 -6.0622i  -4.0000 +5.1962i
  +4.0000 -5.1962i  -2.5000 +6.0622i  +6.5000 -0.8660i  +2.5000 +6.0622i
  +6.5000 +0.8660i  +4.0000 +5.1962i  -6.5000 -4.3301i  -7.0000 -3.4641i
  -0.5000 -7.7942i  -7.0000 +3.4641i  +0.5000 -7.7942i  -6.5000 +4.3301i
  +6.5000 -4.3301i  -0.5000 +7.7942i  +7.0000 -3.4641i  +0.5000 +7.7942i

Phix

Library: Phix/xpGUI

Note MARKSTYLE=DOT added for this task, 1.0.3 has not yet been released, and won't be until work this needs under pwa/p2js is finished.

requires("1.0.3")
include complex.e
constant OMEGA = {-0.5, sqrt(3)*0.5}

// try to replicate Wren sort order for easy comparison
enum NORM, IMAG, REAL, IS_PRIME, ELEN=$
function new_Eisenstein(integer a,b)
    atom {real,imag} = complex_add(complex_mul(OMEGA,b),a)
    integer norm = a*a-a*b+b*b, c = max(abs(a),abs(b))
    bool p = iff(a=0 or b=0 or a=b?is_prime(c) and remainder(c,3)=2
                                  :is_prime(norm))
    return {norm,imag,real,p} -- nb in [NORM..IS_PRIME] order
end function

function Eisenstein()
    sequence eprimes = {}
    for a=-100 to 100 do
        for b=-100 to 100 do
            sequence e = new_Eisenstein(a, b)
            if e[IS_PRIME] then
                eprimes = append(eprimes,e)
            end if
        end for
    end for
    eprimes = sort(eprimes)
    sequence real = vslice(eprimes,REAL),
             imag = vslice(eprimes,IMAG),
             f100 = repeat("",100)
    for i=1 to 100 do -- convert for display
        integer pm = iff(imag[i]>=0?'+':'-')
        f100[i] = sprintf("%7.4f %c%7.4fi",{real[i],pm,abs(imag[i])})
    end for
    return {f100,real,imag}
end function

sequence {f100,real,imag} = Eisenstein()
printf(1,"First 100 Eisenstein primes nearest zero:\n%s\n",join_by(f100,1,4,"  "))

include xpGUI.e
function get_data(gdx graph) return {{real,imag}} end function
gdx graph = gGraph(get_data,"XMIN=-150,XMAX=150,YMIN=-100,YMAX=100"),
    dlg = gDialog(graph,"Eisenstein primes","SIZE=392x290")
gSetAttribute(graph,"GTITLE","with norm <= 100  (%d points)",{length(real)})
gSetAttributes(graph,"XTICK=50,YTICK=25,MARKSTYLE=DOT,GRID=NO")
gShow(dlg)
gMainLoop()

Output same as Wren.

Raku

my \ω = exp 2i × π/3;

sub norm    (@p) { @p[0]² - @p[0]×@p[1] + @p[1]² }
sub display (@p) { (@p[0] + ω×@p[1]).reals».fmt('%+8.4f').join ~ 'i' }

my @E = gather (-10..10 X -10..10).map: -> (\a,\b) {
    take (a,b) if 0 == a|b || a == b ?? (.is-prime and 2 == $_ mod 3 given (a,b)».abs.max) !! norm((a,b)).is-prime
}

(@E.sort: *.&norm).head(100).map(*.&display).batch(4).join("\n").say;

Output:

 -1.5000 -0.8660i  -0.0000 -1.7321i  -1.5000 +0.8660i  +1.5000 -0.8660i
 +0.0000 +1.7321i  +1.5000 +0.8660i  -1.0000 -1.7321i  -2.0000 +0.0000i
 +1.0000 -1.7321i  -1.0000 +1.7321i  +2.0000 +0.0000i  +1.0000 +1.7321i
 -2.0000 -1.7321i  -2.5000 -0.8660i  -0.5000 -2.5981i  -2.5000 +0.8660i
 +0.5000 -2.5981i  -2.0000 +1.7321i  +2.0000 -1.7321i  -0.5000 +2.5981i
 +2.5000 -0.8660i  +0.5000 +2.5981i  +2.5000 +0.8660i  +2.0000 +1.7321i
 -2.5000 -2.5981i  -3.5000 -0.8660i  -1.0000 -3.4641i  -3.5000 +0.8660i
 +1.0000 -3.4641i  -2.5000 +2.5981i  +2.5000 -2.5981i  -1.0000 +3.4641i
 +3.5000 -0.8660i  +1.0000 +3.4641i  +3.5000 +0.8660i  +2.5000 +2.5981i
 -3.5000 -2.5981i  -4.0000 -1.7321i  -0.5000 -4.3301i  -4.0000 +1.7321i
 +0.5000 -4.3301i  -3.5000 +2.5981i  +3.5000 -2.5981i  -0.5000 +4.3301i
 +4.0000 -1.7321i  +0.5000 +4.3301i  +4.0000 +1.7321i  +3.5000 +2.5981i
 -2.5000 -4.3301i  -5.0000 +0.0000i  +2.5000 -4.3301i  -2.5000 +4.3301i
 +5.0000 +0.0000i  +2.5000 +4.3301i  -3.5000 -4.3301i  -5.5000 -0.8660i
 -2.0000 -5.1962i  -5.5000 +0.8660i  +2.0000 -5.1962i  -3.5000 +4.3301i
 +3.5000 -4.3301i  -2.0000 +5.1962i  +5.5000 -0.8660i  +2.0000 +5.1962i
 +5.5000 +0.8660i  +3.5000 +4.3301i  -5.0000 -3.4641i  -5.5000 -2.5981i
 -0.5000 -6.0622i  -5.5000 +2.5981i  +0.5000 -6.0622i  -5.0000 +3.4641i
 +5.0000 -3.4641i  -0.5000 +6.0622i  +5.5000 -2.5981i  +0.5000 +6.0622i
 +5.5000 +2.5981i  +5.0000 +3.4641i  -4.0000 -5.1962i  -6.5000 -0.8660i
 -2.5000 -6.0622i  -6.5000 +0.8660i  +2.5000 -6.0622i  -4.0000 +5.1962i
 +4.0000 -5.1962i  -2.5000 +6.0622i  +6.5000 -0.8660i  +2.5000 +6.0622i
 +6.5000 +0.8660i  +4.0000 +5.1962i  -6.5000 -4.3301i  -7.0000 -3.4641i
 -0.5000 -7.7942i  -7.0000 +3.4641i  +0.5000 -7.7942i  -6.5000 +4.3301i
 +6.5000 -4.3301i  -0.5000 +7.7942i  +7.0000 -3.4641i  +0.5000 +7.7942i

Sidef

Translation of: Raku

class Eisenstein(a, b, w = (-1 + sqrt(3).i)/2) {
    method norm {
        a**2 - a*b + b**2
    }

    method to_s {
        sprintf('%+8.4f%+8.4fi', reals(a + b*w))
    }
}

var E = []

for e in (-10..10 ~X -10..10 -> map_2d {|x,y| Eisenstein(x,y) }) {
    var c = [e.a,e.b].map{.abs}.max
    if (
        ((0 ~~ [e.a, e.b]) || (e.a == e.b)) ?
        (c.is_congruent(2,3) && c.is_prime) : e.norm.is_prime
    ) {
        E << e
    }
}

E.sort_by { .norm }.first(100).slices(4).each {|s|
    say s.join('  ')
}

Output:

 -1.5000 -0.8660i   +0.0000 -1.7321i   -1.5000 +0.8660i   +1.5000 -0.8660i
 +0.0000 +1.7321i   +1.5000 +0.8660i   -1.0000 -1.7321i   -2.0000 +0.0000i
 +1.0000 -1.7321i   -1.0000 +1.7321i   +2.0000 +0.0000i   +1.0000 +1.7321i
 -2.0000 -1.7321i   -2.5000 -0.8660i   -0.5000 -2.5981i   -2.5000 +0.8660i
 +0.5000 -2.5981i   -2.0000 +1.7321i   +2.0000 -1.7321i   -0.5000 +2.5981i
 +2.5000 -0.8660i   +0.5000 +2.5981i   +2.5000 +0.8660i   +2.0000 +1.7321i
 -2.5000 -2.5981i   -3.5000 -0.8660i   -1.0000 -3.4641i   -3.5000 +0.8660i
 +1.0000 -3.4641i   -2.5000 +2.5981i   +2.5000 -2.5981i   -1.0000 +3.4641i
 +3.5000 -0.8660i   +1.0000 +3.4641i   +3.5000 +0.8660i   +2.5000 +2.5981i
 -3.5000 -2.5981i   -4.0000 -1.7321i   -0.5000 -4.3301i   -4.0000 +1.7321i
 +0.5000 -4.3301i   -3.5000 +2.5981i   +3.5000 -2.5981i   -0.5000 +4.3301i
 +4.0000 -1.7321i   +0.5000 +4.3301i   +4.0000 +1.7321i   +3.5000 +2.5981i
 -2.5000 -4.3301i   -5.0000 +0.0000i   +2.5000 -4.3301i   -2.5000 +4.3301i
 +5.0000 +0.0000i   +2.5000 +4.3301i   -3.5000 -4.3301i   -5.5000 -0.8660i
 -2.0000 -5.1962i   -5.5000 +0.8660i   +2.0000 -5.1962i   -3.5000 +4.3301i
 +3.5000 -4.3301i   -2.0000 +5.1962i   +5.5000 -0.8660i   +2.0000 +5.1962i
 +5.5000 +0.8660i   +3.5000 +4.3301i   -5.0000 -3.4641i   -5.5000 -2.5981i
 -0.5000 -6.0622i   -5.5000 +2.5981i   +0.5000 -6.0622i   -5.0000 +3.4641i
 +5.0000 -3.4641i   -0.5000 +6.0622i   +5.5000 -2.5981i   +0.5000 +6.0622i
 +5.5000 +2.5981i   +5.0000 +3.4641i   -4.0000 -5.1962i   -6.5000 -0.8660i
 -2.5000 -6.0622i   -6.5000 +0.8660i   +2.5000 -6.0622i   -4.0000 +5.1962i
 +4.0000 -5.1962i   -2.5000 +6.0622i   +6.5000 -0.8660i   +2.5000 +6.0622i
 +6.5000 +0.8660i   +4.0000 +5.1962i   -6.5000 -4.3301i   -7.0000 -3.4641i
 -0.5000 -7.7942i   -7.0000 +3.4641i   +0.5000 -7.7942i   -6.5000 +4.3301i
 +6.5000 -4.3301i   -0.5000 +7.7942i   +7.0000 -3.4641i   +0.5000 +7.7942i

Wren

Library: DOME

Library: Wren-plot

Library: Wren-iterate

Library: Wren-complex

Library: Wren-math

Library: Wren-fmt

import "dome" for Window
import "graphics" for Canvas, Color
import "./plot" for Axes
import "./iterate" for Stepped
import "./complex" for Complex
import "./math2" for Math, Int
import "./fmt" for Fmt

var OMEGA = Complex.new(-0.5, 3.sqrt * 0.5)

class Eisenstein {
    construct new(a, b) {
        _a = a
        _b = b
        _n = OMEGA * b + a
    }

    a { _a }
    b { _b }
    n { _n }

    real { _n.real }
    imag { _n.imag }
    norm { _a *_a - _a * _b + _b * _b }

    isPrime {
        if (_a == 0 || _b == 0 || _a == _b) {
            var c = Math.max(_a.abs, _b.abs)
            return Int.isPrime(c) && c % 3 == 2
        }
        return Int.isPrime(norm)
    }

    toString { _n.toString }
}

var eprimes = []
for (a in -100..100) {
    for (b in -100..100) {
        var e = Eisenstein.new(a, b)
        if (e.isPrime) eprimes.add(e)
    }
}

// try to replicate Julia sort order for easy comparison
eprimes.sort { |e1, e2|
    if (e1.norm < e2.norm) return true
    if (e1.norm == e2.norm) {
        if (e1.imag < e2.imag) return true
        if (e1.imag == e2.imag) return e1.real < e2.real
        return false
    }
    return false
}

// convert to Complex numbers for easy display
eprimes = eprimes.map { |e| e.n }

// display first 100 to terminal
System.print("First 100 Eisenstein primes nearest zero:")
Fmt.tprint("$ 6.4z ", eprimes.take(100), 4)

// generate points for the plot
var Pts = eprimes.map { |e| [e.real, e.imag] }.toList

class Main {
    construct new() {
        Window.title = "Eisenstein primes with norm <= 100  (%(Pts.count) points)"
        Canvas.resize(1000, 600)
        Window.resize(1000, 600)
        Canvas.cls(Color.white)
        var axes = Axes.new(100, 500, 800, 400, -160..160, -100..100)
        axes.draw(Color.black, 2)
        var xMarks = Stepped.new(-150..150, 50)
        var yMarks = Stepped.new(-75..75, 25)
        axes.mark(xMarks, yMarks, Color.black, 2)
        axes.label(xMarks, yMarks, Color.black, 2, Color.black)
        axes.plot(Pts, Color.black, "·") // uses interpunct character 0xb7
    }

    init() {}

    update() {}

    draw(alpha) {}
}

var Game = Main.new()

Output:

Terminal output:

First 100 Eisenstein primes nearest zero:
 0.0000 - 1.7321i  -1.5000 - 0.8660i   1.5000 - 0.8660i  -1.5000 + 0.8660i 
 1.5000 + 0.8660i   0.0000 + 1.7321i  -1.0000 - 1.7321i   1.0000 - 1.7321i 
-2.0000 + 0.0000i   2.0000 + 0.0000i  -1.0000 + 1.7321i   1.0000 + 1.7321i 
-0.5000 - 2.5981i   0.5000 - 2.5981i  -2.0000 - 1.7321i   2.0000 - 1.7321i 
-2.5000 - 0.8660i   2.5000 - 0.8660i  -2.5000 + 0.8660i   2.5000 + 0.8660i 
-2.0000 + 1.7321i   2.0000 + 1.7321i  -0.5000 + 2.5981i   0.5000 + 2.5981i 
-1.0000 - 3.4641i   1.0000 - 3.4641i  -2.5000 - 2.5981i   2.5000 - 2.5981i 
-3.5000 - 0.8660i   3.5000 - 0.8660i  -3.5000 + 0.8660i   3.5000 + 0.8660i 
-2.5000 + 2.5981i   2.5000 + 2.5981i  -1.0000 + 3.4641i   1.0000 + 3.4641i 
-0.5000 - 4.3301i   0.5000 - 4.3301i  -3.5000 - 2.5981i   3.5000 - 2.5981i 
-4.0000 - 1.7321i   4.0000 - 1.7321i  -4.0000 + 1.7321i   4.0000 + 1.7321i 
-3.5000 + 2.5981i   3.5000 + 2.5981i  -0.5000 + 4.3301i   0.5000 + 4.3301i 
-2.5000 - 4.3301i   2.5000 - 4.3301i  -5.0000 + 0.0000i   5.0000 + 0.0000i 
-2.5000 + 4.3301i   2.5000 + 4.3301i  -2.0000 - 5.1962i   2.0000 - 5.1962i 
-3.5000 - 4.3301i   3.5000 - 4.3301i  -5.5000 - 0.8660i   5.5000 - 0.8660i 
-5.5000 + 0.8660i   5.5000 + 0.8660i  -3.5000 + 4.3301i   3.5000 + 4.3301i 
-2.0000 + 5.1962i   2.0000 + 5.1962i  -0.5000 - 6.0622i   0.5000 - 6.0622i 
-5.0000 - 3.4641i   5.0000 - 3.4641i  -5.5000 - 2.5981i   5.5000 - 2.5981i 
-5.5000 + 2.5981i   5.5000 + 2.5981i  -5.0000 + 3.4641i   5.0000 + 3.4641i 
-0.5000 + 6.0622i   0.5000 + 6.0622i  -2.5000 - 6.0622i   2.5000 - 6.0622i 
-4.0000 - 5.1962i   4.0000 - 5.1962i  -6.5000 - 0.8660i   6.5000 - 0.8660i 
-6.5000 + 0.8660i   6.5000 + 0.8660i  -4.0000 + 5.1962i   4.0000 + 5.1962i 
-2.5000 + 6.0622i   2.5000 + 6.0622i  -0.5000 - 7.7942i   0.5000 - 7.7942i 
-6.5000 - 4.3301i   6.5000 - 4.3301i  -7.0000 - 3.4641i   7.0000 - 3.4641i 
-7.0000 + 3.4641i   7.0000 + 3.4641i  -6.5000 + 4.3301i   6.5000 + 4.3301i