Gaussian primes
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 its norm is a prime number, or 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, then the additive inverse b² + a² is also prime, as are the complex conjugates and multiplicative inverses 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
J
Implementation: <lang J>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</lang>
Plot of gaussian primes up to radius 50: <lang J> 1j1#"1'#' (<"1]50++.(#~ isgpri * 50>:|) ,j./~i:100)} '+' (<50 50)} '|' 50}"1 '-' 50} 100 100$' '
| # # | # # # | # # # # # # # # # # # # | # # # # # # # # | # # # # # # # # | # # # # # # # # # # # # # # # # # # | # # # # # # # # # | # # # # # # # # # # | # # # # # # # # | # # # # # # # # # | # # # # # # # # # # # # # | # # # # # # # # # # | # # # # # # # # # # # # | # # # # # # # # # # # # # # # | # # # # # # # # # # # # | # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # | # # # # # # # # # # # # # # | # # # # # # # # # # # | # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # #
- - - # - - - # - - - - - - - - - - - # - - - - - - - # - - - # - - - - - - - # - - - # - - - # - - + - - # - - - # - - - # - - - - - - - # - - - # - - - - - - - # - - - - - - - - - - - # - - - # - -
# # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # # | # # # # # # # # # # # | # # # # # # # # # # # # # # | # # # # # # # # # # # # # # | # # # # # # # # # # # # # # # # # # # # # # # # # # # # # | # # # # # # # # # # # # # | # # # # # # # # # # # # | # # # # # # # # # # # # # # # | # # # # # # # # # # # # | # # # # # # # # # # | # # # # # # # # # # # # # | # # # # # # # # # | # # # # # # # # | # # # # # # # # # # | # # # # # # # # # | # # # # # # # # # # # # # # # # # # | # # # # # # # # | # # # # # # # # | # # # # # # # # # # # # | # # # | # #
</lang>
Gaussian primes less than radius 10 (sorted by radius):<lang J> 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</lang>
Julia
<lang ruby>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 $lim of the origin on the complex plane:") foreach(p -> print(lpad(p[2], 10), p[1] % 10 == 0 ? "\n" : ""), enumerate(gprimes)) # print scatter(gprimes) # plot
end
testgaussprimes()
</lang>
- Output:
Gaussian primes within 10 of the origin on the complex plane: 1 + 1im 1 - 1im -1 - 1im -1 + 1im 1 + 2im -2 + 1im 2 + 1im 2 - 1im -2 - 1im 1 - 2im -1 - 2im -1 + 2im 3 + 0im -3 + 0im 0 - 3im 0 + 3im -3 - 2im -2 + 3im 3 + 2im 3 - 2im -2 - 3im 2 + 3im 2 - 3im -3 + 2im 4 + 1im 4 - 1im -1 + 4im -4 - 1im -4 + 1im -1 - 4im 1 - 4im 1 + 4im 5 - 2im 2 + 5im -5 + 2im -5 - 2im 5 + 2im -2 + 5im 2 - 5im -2 - 5im 1 - 6im -6 + 1im 6 + 1im -6 - 1im -1 - 6im -1 + 6im 1 + 6im 6 - 1im -4 + 5im 5 + 4im -5 + 4im 4 + 5im 5 - 4im -5 - 4im 4 - 5im -4 - 5im 0 + 7im -7 + 0im 0 - 7im 7 + 0im 7 + 2im -2 + 7im -2 - 7im 2 - 7im 2 + 7im 7 - 2im -7 - 2im -7 + 2im 6 - 5im -6 - 5im 5 + 6im -5 - 6im 5 - 6im -6 + 5im -5 + 6im 6 + 5im 3 + 8im -8 + 3im 8 + 3im -3 + 8im -8 - 3im 8 - 3im 3 - 8im -3 - 8im 8 + 5im -5 - 8im -5 + 8im 5 - 8im -8 + 5im -8 - 5im 8 - 5im 5 + 8im -4 + 9im -4 - 9im 9 + 4im -9 + 4im 9 - 4im -9 - 4im 4 - 9im 4 + 9im
Perl
<lang perl>#!/usr/bin/perl
use strict; # https://rosettacode.org/wiki/Gaussian_primes use warnings; use ntheory qw( is_prime );
my ($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; is_prime( $norm ) or ($norm - 3) % 4 == 0 or next; push @primes, "$A ${B}i"; substr $plot, ($B + $size + 1) * (2 * $size + 2) + $A + $size + 1, 1, 'X'; } } return $plot, @primes; }</lang>
- Output:
Primes within 10 -9 -4i, -9 4i, -8 -5i, -8 -3i, -8 3i, -8 5i, -7 -2i, -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 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, 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 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 2i, 8 -5i, 8 -3i, 8 3i, 8 5i, 9 -4i, 9 4i Plot within 50 X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X XX XX X X X X X X X X X X X X X X X X X X X X XX XX X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X
Phix
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
<lang python> 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])
</lang>
- 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
Plotting the points up to a radius of 150. <lang perl6>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 ],
);</lang>
- 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
Plots the points up to a radius of 150 to produce a similar image to the Raku example. <lang ecmascript>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()</lang>
- 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)