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Gaussian primes

From Rosetta Code
Gaussian primes is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

A Gaussian Integer is a complex number such that its real and imaginary parts are both integers.

   a + bi where a and b are integers and i is √-1.

The norm of a Gaussian integer is its product with its conjugate.

   N(a + bi) = (a + bi)(a − bi) = a² + b²


A Gaussian integer is a Gaussian prime if and only if either: both a and b are non-zero and its norm is a prime number, or, one of a or b is zero and it is the product of a unit (±1, ±i) and a prime integer of the form 4n + 3.

Prime integers that are not of the form 4n + 3 can be factored into a Gaussian integer and its complex conjugate so are not a Gaussian prime.

   E.G. 5 = (2 + i)(2 − i) So 5 is not a Gaussian prime

Gaussian primes are octogonally symmetrical on a real / imaginary Cartesian field. If a particular complex norm a² + b² is prime, since addition is commutative, b² + a² is also prime, as are the complex conjugates and negated instances of both.


Task

Find and show, here on this page, the Gaussian primes with a norm of less than 100, (within a radius of 10 from the origin 0 + 0i on a complex plane.)

Plot the points corresponding to the Gaussian primes on a Cartesian real / imaginary plane at least up to a radius of 50.


See also


F#[edit]

This task uses Extensible Prime Generator (F#)

 
// Gaussian primes. Nigel Galloway: July 29th., 2022
let isGP=function (n,0)|(0,n)->let n=abs n in n%4=3 && isPrime n |(n,g)->isPrime(n*n+g*g)
Seq.allPairs [-9..9] [-9..9]|>Seq.filter isGP|>Seq.iter(fun(n,g)->printf $"""%d{n}%s{match g with 0->"" |g->sprintf $"%+d{g}i"} """); printfn ""
 
Output:
-9-4i -9+4i -8-7i -8-5i -8-3i -8+3i -8+5i -8+7i -7-8i -7-2i -7 -7+2i -7+8i -6-5i -6-1i -6+1i -6+5i -5-8i -5-6i -5-4i -5-2i -5+2i -5+4i -5+6i -5+8i -4-9i -4-5i -4-1i -4+1i -4+5i -4+9i -3-8i -3-2i -3 -3+2i -3+8i -2-7i -2-5i -2-3i -2-1i -2+1i -2+3i -2+5i -2+7i -1-6i -1-4i -1-2i -1-1i -1+1i -1+2i -1+4i -1+6i 0-7i 0-3i 0+3i 0+7i 1-6i 1-4i 1-2i 1-1i 1+1i 1+2i 1+4i 1+6i 2-7i 2-5i 2-3i 2-1i 2+1i 2+3i 2+5i 2+7i 3-8i 3-2i 3 3+2i 3+8i 4-9i 4-5i 4-1i 4+1i 4+5i 4+9i 5-8i 5-6i 5-4i 5-2i 5+2i 5+4i 5+6i 5+8i 6-5i 6-1i 6+1i 6+5i 7-8i 7-2i 7 7+2i 7+8i 8-7i 8-5i 8-3i 8+3i 8+5i 8+7i 9-4i 9+4i

J[edit]

Implementation:
isgpri=: {{
if. 1 p: (*+) y do. 1 return. end.
int=. |(+.y)-.0
if. 1=#int do. {.(1 p: int) * 3=4|int else. 0 end.
}}"0

Online plot of gaussian primes up to radius 100. (Hit "Run" in the upper right-hand corner.)

Plot of gaussian primes up to radius 50:
   1j1#"1'#' (<"1]50++.(#~ isgpri * 50>:|) ,j./~i:100)} '+' (<50 50)} '|' 50}"1 '-' 50} 100 100$' '
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Gaussian primes less than radius 10 (sorted by radius):
   10 10$(/: |)(#~ isgpri * 10>|) ,j./~i:10
_1j_1 _1j1 1j_1 1j1 _2j_1 _2j1 _1j_2 _1j2 1j_2 1j2
2j_1 2j1 _3 0j_3 0j3 3 _3j_2 _3j2 _2j_3 _2j3
2j_3 2j3 3j_2 3j2 _4j_1 _4j1 _1j_4 _1j4 1j_4 1j4
4j_1 4j1 _5j_2 _5j2 _2j_5 _2j5 2j_5 2j5 5j_2 5j2
_6j_1 _6j1 _1j_6 _1j6 1j_6 1j6 6j_1 6j1 _5j_4 _5j4
_4j_5 _4j5 4j_5 4j5 5j_4 5j4 _7 0j_7 0j7 7
_7j_2 _7j2 _2j_7 _2j7 2j_7 2j7 7j_2 7j2 _6j_5 _6j5
_5j_6 _5j6 5j_6 5j6 6j_5 6j5 _8j_3 _8j3 _3j_8 _3j8
3j_8 3j8 8j_3 8j3 _8j_5 _8j5 _5j_8 _5j8 5j_8 5j8
8j_5 8j5 _9j_4 _9j4 _4j_9 _4j9 4j_9 4j9 9j_4 9j4

Julia[edit]

using LinearAlgebra
using Plots
using Primes
 
"""
function isGaussianprime(n::Complex{T}) where T <: Integer
 
A Gaussian prime is a non-unit Gaussian integer m + ni divisible only by its associates and by the units
1, i, -1, -i and by no other Gaussian integers.
 
The Gaussian primes fall into one of three categories:
 
Gaussian integers with imaginary part zero and a prime real part m with |m| a real prime satisfying |m| = 3 mod 4
Gaussian integers with real part zero and an imaginary part n with |n| real prime satisfying |n| = 3 mod 4
Gaussian integers having both real and imaginary parts, and its complex norm (square of algebraic norm) is a real prime number
"
""
function isGaussianprime(n::Complex{T}) where T <: Integer
r, c = abs(real(n)), abs(imag(n))
return isprime(r * r + c * c) || c == 0 && isprime(r) && (r - 3) % 4 == 0 || r == 0 && isprime(c) && (c - 3) % 4 == 0
end
 
function testgaussprimes(lim = 10)
testvals = map(c -> c[1] + im * c[2], collect(Iterators.product(-lim:lim, -lim:lim)))
gprimes = sort!(filter(c -> isGaussianprime(c) && norm(c) < lim, testvals), by = norm)
println("Gaussian primes within $lim of the origin on the complex plane:")
foreach(p -> print(lpad(p[2], 10), p[1] % 10 == 0 ? "\n" : ""), enumerate(gprimes)) # print
scatter(gprimes) # plot
end
 
testgaussprimes()
 
Output:
Gaussian primes within 10 of the origin on the complex plane:
   1 + 1im   1 - 1im  -1 - 1im  -1 + 1im   1 + 2im  -2 + 1im   2 + 1im   2 - 1im  -2 - 1im   1 - 2im
  -1 - 2im  -1 + 2im   3 + 0im  -3 + 0im   0 - 3im   0 + 3im  -3 - 2im  -2 + 3im   3 + 2im   3 - 2im
  -2 - 3im   2 + 3im   2 - 3im  -3 + 2im   4 + 1im   4 - 1im  -1 + 4im  -4 - 1im  -4 + 1im  -1 - 4im
   1 - 4im   1 + 4im   5 - 2im   2 + 5im  -5 + 2im  -5 - 2im   5 + 2im  -2 + 5im   2 - 5im  -2 - 5im
   1 - 6im  -6 + 1im   6 + 1im  -6 - 1im  -1 - 6im  -1 + 6im   1 + 6im   6 - 1im  -4 + 5im   5 + 4im
  -5 + 4im   4 + 5im   5 - 4im  -5 - 4im   4 - 5im  -4 - 5im   0 + 7im  -7 + 0im   0 - 7im   7 + 0im
   7 + 2im  -2 + 7im  -2 - 7im   2 - 7im   2 + 7im   7 - 2im  -7 - 2im  -7 + 2im   6 - 5im  -6 - 5im
   5 + 6im  -5 - 6im   5 - 6im  -6 + 5im  -5 + 6im   6 + 5im   3 + 8im  -8 + 3im   8 + 3im  -3 + 8im
  -8 - 3im   8 - 3im   3 - 8im  -3 - 8im   8 + 5im  -5 - 8im  -5 + 8im   5 - 8im  -8 + 5im  -8 - 5im
   8 - 5im   5 + 8im  -4 + 9im  -4 - 9im   9 + 4im  -9 + 4im   9 - 4im  -9 - 4im   4 - 9im   4 + 9im

Mathematica/Wolfram Language[edit]

n = 100;
digs = [email protected][If[Norm[i + I j]^2 < n, If[PrimeQ[i + I j, GaussianIntegers -> True], Sow[i + I j]]],
{i,Floor[-Sqrt[n]], Ceiling[Sqrt[n]]}, {j, Floor[-Sqrt[n]], Ceiling[Sqrt[n]]}
];
Multicolumn[digs[[2, 1]], Appearance -> "Horizontal"]
 
n = 50^2;
digs = Table[If[Norm[i + I j]^2 < n, If[PrimeQ[i + I j, GaussianIntegers -> True], "*", " "], " "],
{i,Floor[-Sqrt[n]], Ceiling[Sqrt[n]]}, {j, Floor[-Sqrt[n]], Ceiling[Sqrt[n]]}
];
digs //= Map[StringJoin];
digs //= StringRiffle[#, "\n"] &;
digs
Output:
-9-4 I	-9+4 I	-8-5 I	-8-3 I	-8+3 I	-8+5 I	-7-2 I	-7	-7+2 I	-6-5 I
-6-I	-6+I	-6+5 I	-5-8 I	-5-6 I	-5-4 I	-5-2 I	-5+2 I	-5+4 I	-5+6 I
-5+8 I	-4-9 I	-4-5 I	-4-I	-4+I	-4+5 I	-4+9 I	-3-8 I	-3-2 I	-3
-3+2 I	-3+8 I	-2-7 I	-2-5 I	-2-3 I	-2-I	-2+I	-2+3 I	-2+5 I	-2+7 I
-1-6 I	-1-4 I	-1-2 I	-1-I	-1+I	-1+2 I	-1+4 I	-1+6 I	-7 I	-3 I
3 I	7 I	1-6 I	1-4 I	1-2 I	1-I	1+I	1+2 I	1+4 I	1+6 I
2-7 I	2-5 I	2-3 I	2-I	2+I	2+3 I	2+5 I	2+7 I	3-8 I	3-2 I
3	3+2 I	3+8 I	4-9 I	4-5 I	4-I	4+I	4+5 I	4+9 I	5-8 I
5-6 I	5-4 I	5-2 I	5+2 I	5+4 I	5+6 I	5+8 I	6-5 I	6-I	6+I
6+5 I	7-2 I	7	7+2 I	8-5 I	8-3 I	8+3 I	8+5 I	9-4 I	9+4 I


                                                                                                     
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Perl[edit]

Library: ntheory
#!/usr/bin/perl
 
use strict; # https://rosettacode.org/wiki/Gaussian_primes
use warnings;
use ntheory qw( is_prime );
 
my ($plot, @primes) = gaussianprimes(10);
print "Primes within 10\n", join(', ', @primes) =~ s/.{94}\K /\n/gr;
($plot, @primes) = gaussianprimes(50);
print "\n\nPlot within 50\n$plot";
 
sub gaussianprimes
{
my $size = shift;
my $plot = ( ' ' x (2 * $size + 1) . "\n" ) x (2 * $size + 1);
my @primes;
for my $A ( -$size .. $size )
{
my $limit = int sqrt $size**2 - $A**2;
for my $B ( -$limit .. $limit )
{
my $norm = $A**2 + $B**2;
if ( is_prime( $norm )
or ( $A==0 && is_prime(abs $B) && (abs($B)-3)%4 == 0)
or ( $B==0 && is_prime(abs $A) && (abs($A)-3)%4 == 0) )
{
push @primes, sprintf("%2d%2di", $A, $B) =~ s/ (\di)/+$1/r;
substr $plot, ($B + $size + 1) * (2 * $size + 2) + $A + $size + 1, 1, 'X';
}
}
}
return $plot, @primes;
}
Output:
Primes within 10
-9-4i,  -9+4i,  -8-5i,  -8-3i,  -8+3i,  -8+5i,  -7-2i,  -7+0i,  -7+2i,  -6-5i,  -6-1i,  -6+1i,
-6+5i,  -5-8i,  -5-6i,  -5-4i,  -5-2i,  -5+2i,  -5+4i,  -5+6i,  -5+8i,  -4-9i,  -4-5i,  -4-1i,
-4+1i,  -4+5i,  -4+9i,  -3-8i,  -3-2i,  -3+0i,  -3+2i,  -3+8i,  -2-7i,  -2-5i,  -2-3i,  -2-1i,
-2+1i,  -2+3i,  -2+5i,  -2+7i,  -1-6i,  -1-4i,  -1-2i,  -1-1i,  -1+1i,  -1+2i,  -1+4i,  -1+6i,
 0-7i,   0-3i,   0+3i,   0+7i,   1-6i,   1-4i,   1-2i,   1-1i,   1+1i,   1+2i,   1+4i,   1+6i,
 2-7i,   2-5i,   2-3i,   2-1i,   2+1i,   2+3i,   2+5i,   2+7i,   3-8i,   3-2i,   3+0i,   3+2i,
 3+8i,   4-9i,   4-5i,   4-1i,   4+1i,   4+5i,   4+9i,   5-8i,   5-6i,   5-4i,   5-2i,   5+2i,
 5+4i,   5+6i,   5+8i,   6-5i,   6-1i,   6+1i,   6+5i,   7-2i,   7+0i,   7+2i,   8-5i,   8-3i,
 8+3i,   8+5i,   9-4i,   9+4i
Plot within 50
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Phix[edit]

Library: Phix/pGUI
Library: Phix/online

You can run this online here.

--
-- demo/rosetta/Gaussian_primes.exw
-- ================================
--
with javascript_semantics

function gaussian_primes(integer radius)
    integer sq_radius = radius*radius
    sequence res = {}
    for i=1 to radius do
        if remainder(i,4)=3 then
            res = append(res,{i*i,i,0})
        end if
        integer i2 = i*i
        for j=i to radius do
            integer r = i2+j*j
            if r>sq_radius then exit end if
            if is_prime(r) then
                res = append(res,{r,j,i})
            end if
        end for
    end for
    res = sort(res)
    return res
end function

include builtins\complex.e
function gpp(integer i, j)
    return pad_head(complex_sprint({i,j}),6)
end function

function g4(integer i,j)
    sequence res = {gpp(i,j)}
    if i!=0 then
        res = append(res,gpp(-i,j))
        if j!=0 then
            res = append(res,gpp(-i,-j))
        end if
    end if
    if j!=0 then
        res = append(res,gpp(i,-j))
    end if
    return res
end function

function reflect(sequence g)
    sequence res = {}
    for p in g do
        integer {n,i,j} = p
        res &= g4(i,j)
        if i!=j then res &= g4(j,i) end if
    end for
    return res
end function

sequence g = gaussian_primes(10)

printf(1,"Gaussian primes with a norm less than 100 sorted by norm:\n%s\n",
         {join_by(reflect(g),1,10," ")})

--g = gaussian_primes(50) -- (radius of 50)
g = gaussian_primes(150) -- (radius of 150)

constant title = "Gaussian primes"
include pGUI.e

Ihandle dlg, canvas
cdCanvas cddbuffer, cdcanvas
integer cx, cy

procedure plot4(integer i,j)
    for im=+1 to -1 by -2 do
        for jm=+1 to -1 by -2 do
            cdCanvasPixel(cddbuffer, cx+im*i, cy+jm*j, CD_YELLOW) 
        end for
    end for
end procedure

function redraw_cb(Ihandle /*ih*/)
    integer {width, height} = IupGetIntInt(canvas, "DRAWSIZE")
    cx = floor(width/2)
    cy = floor(height/2)
    cdCanvasActivate(cddbuffer)
    cdCanvasClear(cddbuffer)
    for p in g do
        integer {n,i,j} = p
        plot4(i,j)
        plot4(j,i)
    end for
    cdCanvasFlush(cddbuffer)
    return IUP_DEFAULT
end function

function map_cb(Ihandle ih)
    cdcanvas = cdCreateCanvas(CD_IUP, ih)
    cddbuffer = cdCreateCanvas(CD_DBUFFER, cdcanvas)
    cdCanvasSetBackground(cddbuffer, CD_BLACK)
    return IUP_DEFAULT
end function

IupOpen()
--canvas = IupCanvas("RASTERSIZE=320x320")
canvas = IupCanvas("RASTERSIZE=340x340")
IupSetCallbacks(canvas, {"MAP_CB", Icallback("map_cb"),
                         "ACTION", Icallback("redraw_cb")})
dlg = IupDialog(canvas, `TITLE="%s",RESIZE=NO`,{title})
IupShow(dlg)
if platform()!=JS then
    IupMainLoop()
    IupClose()
end if

Output same as Raku

Python[edit]

''' python example for task rosettacode.org/wiki/Gaussian_primes '''
 
from matplotlib.pyplot import scatter
from sympy import isprime
from math import isqrt
 
def norm(c):
''' Task complex norm function '''
return c.real * c.real + c.imag * c.imag
 
 
def is_gaussian_prime(n):
'''
is_gaussian_prime(n)
 
A Gaussian prime is a non-unit Gaussian integer m + ni divisible only by its associates and by the units
1, i, -1, -i and by no other Gaussian integers.
 
The Gaussian primes fall into one of three categories:
 
Gaussian integers with imaginary part zero and a prime real part m with |m| a real prime satisfying |m| = 3 mod 4
Gaussian integers with real part zero and an imaginary part n with |n| real prime satisfying |n| = 3 mod 4
Gaussian integers having both real and imaginary parts, and its complex norm (square of algebraic norm) is a real prime number
'''

r, c = int(abs(n.real)), int(abs(n.imag))
return isprime(r * r + c * c) or c == 0 and isprime(r) and (r - 3) % 4 == 0 or r == 0 and isprime(c) and (c - 3) % 4 == 0
 
if __name__ == '__main__':
 
limitsquared = 100
lim = isqrt(limitsquared)
testvals = [complex(r, c) for r in range(-lim, lim) for c in range(-lim, lim)]
gprimes = sorted(filter(lambda c : is_gaussian_prime(c) and norm(c) < limitsquared, testvals), key=norm)
print(f'Gaussian primes within {isqrt(limitsquared)} of the origin on the complex plane:')
for i, c in enumerate(gprimes):
print(str(c).ljust(9), end='\n' if (i +1) % 10 == 0 else '')
scatter([c.real for c in gprimes], [c.imag for c in gprimes])
 
Output:
Gaussian primes within 10 of the origin on the complex plane:
(-1-1j)  (-1+1j)  (1-1j)   (1+1j)   (-2-1j)  (-2+1j)  (-1-2j)  (-1+2j)  (1-2j)   (1+2j)   
(2-1j)   (2+1j)   (-3+0j)  -3j      3j       (3+0j)   (-3-2j)  (-3+2j)  (-2-3j)  (-2+3j)  
(2-3j)   (2+3j)   (3-2j)   (3+2j)   (-4-1j)  (-4+1j)  (-1-4j)  (-1+4j)  (1-4j)   (1+4j)   
(4-1j)   (4+1j)   (-5-2j)  (-5+2j)  (-2-5j)  (-2+5j)  (2-5j)   (2+5j)   (5-2j)   (5+2j)   
(-6-1j)  (-6+1j)  (-1-6j)  (-1+6j)  (1-6j)   (1+6j)   (6-1j)   (6+1j)   (-5-4j)  (-5+4j)  
(-4-5j)  (-4+5j)  (4-5j)   (4+5j)   (5-4j)   (5+4j)   (-7+0j)  -7j      7j       (7+0j)   
(-7-2j)  (-7+2j)  (-2-7j)  (-2+7j)  (2-7j)   (2+7j)   (7-2j)   (7+2j)   (-6-5j)  (-6+5j)  
(-5-6j)  (-5+6j)  (5-6j)   (5+6j)   (6-5j)   (6+5j)   (-8-3j)  (-8+3j)  (-3-8j)  (-3+8j)  
(3-8j)   (3+8j)   (8-3j)   (8+3j)   (-8-5j)  (-8+5j)  (-5-8j)  (-5+8j)  (5-8j)   (5+8j)   
(8-5j)   (8+5j)   (-9-4j)  (-9+4j)  (-4-9j)  (-4+9j)  (4-9j)   (4+9j)   (9-4j)   (9+4j)

Raku[edit]

Plotting the points up to a radius of 150.

use List::Divvy;
 
my @next = { :1x, :1y, :2n },;
 
sub next-interval (Int $int) {
@next.append: (^$int).map: { %( :x($int), :y($_), :n($int² + .²) ) };
@next = |@next.sort: *.<n>;
}
 
my @gaussian = lazy gather {
my $interval = 1;
loop {
my @this = @next.shift;
@this.push: @next.shift while @next and @next[0]<n> == @this[0]<n>;
for @this {
.take if .<n>.is-prime || (!.<y> && .<x>.is-prime && (.<x> - 3) %% 4);
next-interval(++$interval) if $interval == .<x>
}
}
}
 
# Primes within a radius of 10 from origin
say "Gaussian primes with a norm less than 100 sorted by norm:";
say @gaussian.&before(*.<n> > 10²).map( {
my (\i, \j) = .<x y>;
flat ((i,j),(-i,j),(-i,-j),(i,-j),(j,i),(-j,i),(-j,-i),(j,-i)).map: {
.[0] ?? .[1] ?? (sprintf "%d%s%di", .[0], (.[1]0 ?? '+' !! ''), .[1]) !! .[0] !! "{.[1]}i"
}} )».subst('1i', 'i', :g)».fmt("%6s")».unique.flat.batch(10).join: "\n" ;
 
 
# Plot points within a 150 radius
use SVG;
 
my @points = unique flat @gaussian.&before(*.<n> > 150²).map: {
my (\i, \j) = .<x y>;
do for (i,j),(-i,j),(-i,-j),(i,-j),(j,i),(-j,i),(-j,-i),(j,-i) {
:use['xlink:href'=>'#point', 'transform'=>"translate({500 + 3 × .[0]},{500 + 3 × .[1]})"]
}
}
 
'gaussian-primes-raku.svg'.IO.spurt: SVG.serialize(
svg => [
:width<1000>, :height<1000>,
:rect[:width<100%>, :height<100%>, :style<fill:black;>],
:defs[:g[:id<point>, :circle[:0cx, :0cy, :2r, :fill('gold')]]],
|@points
],
);
Output:
Gaussian primes with a norm less than 100 sorted by norm:
   1+i   -1+i   -1-i    1-i    2+i   -2+i   -2-i    2-i   1+2i  -1+2i
 -1-2i   1-2i      3     -3     3i    -3i   3+2i  -3+2i  -3-2i   3-2i
  2+3i  -2+3i  -2-3i   2-3i    4+i   -4+i   -4-i    4-i   1+4i  -1+4i
 -1-4i   1-4i   5+2i  -5+2i  -5-2i   5-2i   2+5i  -2+5i  -2-5i   2-5i
   6+i   -6+i   -6-i    6-i   1+6i  -1+6i  -1-6i   1-6i   5+4i  -5+4i
 -5-4i   5-4i   4+5i  -4+5i  -4-5i   4-5i      7     -7     7i    -7i
  7+2i  -7+2i  -7-2i   7-2i   2+7i  -2+7i  -2-7i   2-7i   6+5i  -6+5i
 -6-5i   6-5i   5+6i  -5+6i  -5-6i   5-6i   8+3i  -8+3i  -8-3i   8-3i
  3+8i  -3+8i  -3-8i   3-8i   8+5i  -8+5i  -8-5i   8-5i   5+8i  -5+8i
 -5-8i   5-8i   9+4i  -9+4i  -9-4i   9-4i   4+9i  -4+9i  -4-9i   4-9i

Off-site SVG image: gaussian-primes-raku.svg

Wren[edit]

Library: DOME
Library: Wren-plot
Library: Wren-complex
Library: Wren-math
Library: wren-fmt

Plots the points up to a radius of 150 to produce a similar image to the Raku example.

import "dome" for Window
import "graphics" for Canvas, Color
import "./plot" for Axes
import "./complex" for Complex
import "./math2" for Int
import "./fmt" for Fmt
 
var norm = Fn.new { |c| c.real * c.real + c.imag * c.imag }
 
var GPrimes = []
var Radius = 150
for (r in -Radius+1...Radius) {
for (i in -Radius+1...Radius) {
if (i == 0) {
var m = r.abs
if (Int.isPrime(m) && (m - 3) % 4 == 0) GPrimes.add(Complex.new(r))
} else if (r == 0) {
var m = i.abs
if (Int.isPrime(m) && (m - 3) % 4 == 0) GPrimes.add(Complex.new(0, i))
} else {
var n = r * r + i * i
if (n < Radius * Radius && Int.isPrime(n)) GPrimes.add(Complex.new(r, i))
}
}
}
 
var gp10 = GPrimes.where { |p| norm.call(p) < 100 }.toList
gp10.sort { |i, j|
var ni = norm.call(i)
var nj = norm.call(j)
if (ni != nj) return ni < nj
if (i.real != j.real) return i.real > j.real
return i.imag > j.imag
}
System.print("Gaussian primes with a norm less than 100 sorted by norm:")
Fmt.tprint("($2.0z) ", gp10, 5)
GPrimes = GPrimes.map { |c| c.toPair }.toList
 
class Main {
construct new() {
Window.title = "Gaussian primes"
Canvas.resize(1000, 1000)
Window.resize(1000, 1000)
Canvas.cls(Color.black)
var axes = Axes.new(100, 900, 800, 800, -Radius..Radius, -Radius..Radius)
axes.plot(GPrimes, Color.yellow, "·")
}
 
init() {}
 
update() {}
 
draw(alpha) {}
}
 
var Game = Main.new()
Output:

Terminal output:

Gaussian primes with a norm less than 100 sorted by norm:
( 1 +  1i)  ( 1 -  1i)  (-1 +  1i)  (-1 -  1i)  ( 2 +  1i)  
( 2 -  1i)  ( 1 +  2i)  ( 1 -  2i)  (-1 +  2i)  (-1 -  2i)  
(-2 +  1i)  (-2 -  1i)  ( 3 +  0i)  ( 0 +  3i)  ( 0 -  3i)  
(-3 +  0i)  ( 3 +  2i)  ( 3 -  2i)  ( 2 +  3i)  ( 2 -  3i)  
(-2 +  3i)  (-2 -  3i)  (-3 +  2i)  (-3 -  2i)  ( 4 +  1i)  
( 4 -  1i)  ( 1 +  4i)  ( 1 -  4i)  (-1 +  4i)  (-1 -  4i)  
(-4 +  1i)  (-4 -  1i)  ( 5 +  2i)  ( 5 -  2i)  ( 2 +  5i)  
( 2 -  5i)  (-2 +  5i)  (-2 -  5i)  (-5 +  2i)  (-5 -  2i)  
( 6 +  1i)  ( 6 -  1i)  ( 1 +  6i)  ( 1 -  6i)  (-1 +  6i)  
(-1 -  6i)  (-6 +  1i)  (-6 -  1i)  ( 5 +  4i)  ( 5 -  4i)  
( 4 +  5i)  ( 4 -  5i)  (-4 +  5i)  (-4 -  5i)  (-5 +  4i)  
(-5 -  4i)  ( 7 +  0i)  ( 0 +  7i)  ( 0 -  7i)  (-7 +  0i)  
( 7 +  2i)  ( 7 -  2i)  ( 2 +  7i)  ( 2 -  7i)  (-2 +  7i)  
(-2 -  7i)  (-7 +  2i)  (-7 -  2i)  ( 6 +  5i)  ( 6 -  5i)  
( 5 +  6i)  ( 5 -  6i)  (-5 +  6i)  (-5 -  6i)  (-6 +  5i)  
(-6 -  5i)  ( 8 +  3i)  ( 8 -  3i)  ( 3 +  8i)  ( 3 -  8i)  
(-3 +  8i)  (-3 -  8i)  (-8 +  3i)  (-8 -  3i)  ( 8 +  5i)  
( 8 -  5i)  ( 5 +  8i)  ( 5 -  8i)  (-5 +  8i)  (-5 -  8i)  
(-8 +  5i)  (-8 -  5i)  ( 9 +  4i)  ( 9 -  4i)  ( 4 +  9i)  
( 4 -  9i)  (-4 +  9i)  (-4 -  9i)  (-9 +  4i)  (-9 -  4i)