Gaussian primes
A Gaussian Integer is a complex number such that its real and imaginary parts are both integers.
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 + 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
11l
F is_prime(a)
I a == 2
R 1B
I a < 2 | a % 2 == 0
R 0B
L(i) (3 .. Int(sqrt(a))).step(2)
I a % i == 0
R 0B
R 1B
F is_gaussian_prime(n)
V (r, c) = (Int(abs(n.real)), Int(abs(n.imag)))
R is_prime(r * r + c * c) | (c == 0 & is_prime(r) & (r - 3) % 4 == 0) | (r == 0 & is_prime(c) & (c - 3) % 4 == 0)
F norm(c)
R c.real * c.real + c.imag * c.imag
V limitsquared = 100
V lim = Int(sqrt(limitsquared))
V testvals = multiloop((-lim .< lim), (-lim .< lim), (r, c) -> Complex(r, c))
testvals .= filter(c -> is_gaussian_prime(c) & norm(c) < :limitsquared)
V gprimes = sorted(enumerate(testvals).map((i, c) -> (c, i)), key' c -> (norm(c[0]), c[1]))
print(‘Gaussian primes within ’lim‘ of the origin on the complex plane:’)
L(c) gprimes
print(String(c[0]).ljust(9), end' I (L.index + 1) % 10 == 0 {"\n"} E ‘’)- Output:
Gaussian primes within 10 of the origin on the complex plane: -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 -3i 3i 3 -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 -7i 7i 7 -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
ALGOL 68
The ALGOL 68-primes source is on a page in Rosetta Code - see the link above.
BEGIN # find and plot Gaussian primes: complex numbers with integer real and #
# imaginary parts where: #
# either the real and imaginary parts are non 0 and the norm #
# (sum of squares of the real and imaginary parts) is prime #
# or one of the real/imaginary parts is zero and the other is #
# +/- a prime of the form 4n + 3 #
PR read "primes.incl.a68" PR # include prime utilities #
[]BOOL primes = PRIMESIEVE 10 000; # sieve the primes to 10 000 #
# applies gp to the Gaussian primes with within radius of 0+0i #
PROC process gaussian primes = ( INT radius, PROC(INT,INT)VOID gp )VOID:
BEGIN
INT r2 = radius * radius;
FOR rp FROM - radius TO radius DO
FOR ip FROM - radius TO radius DO
IF INT norm = ( rp * rp ) + ( ip * ip );
norm < r2
THEN
IF rp /= 0 AND ip /= 0 THEN
IF primes[ norm ] THEN gp( rp, ip ) FI
ELIF rp /= 0 OR ip /= 0 THEN
IF INT abs sum parts = ABS ( rp + ip );
primes[ abs sum parts ]
AND ( abs sum parts - 3 ) MOD 4 = 0
THEN
gp( rp, ip )
FI
FI
FI
OD
OD
END # process gaussian primes # ;
# show Gaussian primes within a radius (root of the norm) of 10 of 0+0i #
INT gp count := 0;
process gaussian primes( 10
, ( INT rp, ip )VOID:
BEGIN
print( ( " "
, IF rp > 0 THEN " " ELSE "-" FI, whole( ABS rp, 0 )
, IF ip < 0 THEN "-" ELSE "+" FI, whole( ABS ip, 0 )
, "i"
)
),
IF ( gp count +:= 1 ) MOD 12 = 0 THEN print( ( newline ) ) FI
END
);
# plot the Gaussian primes within a radius of 50 of the origin (0+0i) #
[ -50 : 50, -50 : 50 ]CHAR plot;
FOR i FROM 1 LWB plot TO 1 UPB plot DO
FOR j FROM 2 LWB plot TO 2 UPB plot DO
plot[ i, j ] := IF i = 0 THEN "-" ELSE " " FI
OD;
plot[ i, 0 ] := "|"
OD;
plot[ 0, 0 ] := "+";
process gaussian primes( 50, ( INT rp, ip )VOID: plot[ rp, ip ] := "*" );
print( ( newline, newline ) );
FOR i FROM 1 LWB plot TO 1 UPB plot DO
FOR j FROM 2 LWB plot TO 2 UPB plot DO
print( ( " ", plot[ i, j ] ) )
OD;
print( ( newline ) )
OD
END- Output:
-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
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F#
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 $"%+d{g}i"} """); printfn ""
- Output:
-9-4i -9+4i -8-7i -8-5i -8-3i -8+3i -8+5i -8+7i -7-8i -7-2i -7 -7+2i -7+8i -6-5i -6-1i -6+1i -6+5i -5-8i -5-6i -5-4i -5-2i -5+2i -5+4i -5+6i -5+8i -4-9i -4-5i -4-1i -4+1i -4+5i -4+9i -3-8i -3-2i -3 -3+2i -3+8i -2-7i -2-5i -2-3i -2-1i -2+1i -2+3i -2+5i -2+7i -1-6i -1-4i -1-2i -1-1i -1+1i -1+2i -1+4i -1+6i 0-7i 0-3i 0+3i 0+7i 1-6i 1-4i 1-2i 1-1i 1+1i 1+2i 1+4i 1+6i 2-7i 2-5i 2-3i 2-1i 2+1i 2+3i 2+5i 2+7i 3-8i 3-2i 3 3+2i 3+8i 4-9i 4-5i 4-1i 4+1i 4+5i 4+9i 5-8i 5-6i 5-4i 5-2i 5+2i 5+4i 5+6i 5+8i 6-5i 6-1i 6+1i 6+5i 7-8i 7-2i 7 7+2i 7+8i 8-7i 8-5i 8-3i 8+3i 8+5i 8+7i 9-4i 9+4i
J
Implementation:
isgpri=: {{
if. 1 p: (*+) y do. 1 return. end.
int=. |(+.y)-.0
if. 1=#int do. {.(1 p: int) * 3=4|int else. 0 end.
}}"0
Online plot of gaussian primes up to radius 100. (Hit "Run" in the upper right-hand corner.)
Plot of gaussian primes up to radius 50:
1j1#"1'#' (<"1]50++.(#~ isgpri * 50>:|) ,j./~i:100)} '+' (<50 50)} '|' 50}"1 '-' 50} 100 100$' '
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Gaussian primes less than radius 10 (sorted by radius):
10 10$(/: |)(#~ isgpri * 10>|) ,j./~i:10
_1j_1 _1j1 1j_1 1j1 _2j_1 _2j1 _1j_2 _1j2 1j_2 1j2
2j_1 2j1 _3 0j_3 0j3 3 _3j_2 _3j2 _2j_3 _2j3
2j_3 2j3 3j_2 3j2 _4j_1 _4j1 _1j_4 _1j4 1j_4 1j4
4j_1 4j1 _5j_2 _5j2 _2j_5 _2j5 2j_5 2j5 5j_2 5j2
_6j_1 _6j1 _1j_6 _1j6 1j_6 1j6 6j_1 6j1 _5j_4 _5j4
_4j_5 _4j5 4j_5 4j5 5j_4 5j4 _7 0j_7 0j7 7
_7j_2 _7j2 _2j_7 _2j7 2j_7 2j7 7j_2 7j2 _6j_5 _6j5
_5j_6 _5j6 5j_6 5j6 6j_5 6j5 _8j_3 _8j3 _3j_8 _3j8
3j_8 3j8 8j_3 8j3 _8j_5 _8j5 _5j_8 _5j8 5j_8 5j8
8j_5 8j5 _9j_4 _9j4 _4j_9 _4j9 4j_9 4j9 9j_4 9j4
jq
In this entry, complex numbers are uniformly represented by a two-element array, [real, imaginary], and the function Gprimes(r) emits the Gaussian primes with norm less than r. For the first task, they are displayed in groups, with each group having the same real part.
To use gnuplot to create a plot of the Gaussian primes with norm less than 100, run the program with `plot(100)` (using the jq options `-nr`) to create a file, say gaussian.dat, and then run the gnuplot command:
plot "gaussian.dat" with dots
Preliminaries
### Complex numbers
# For clarity:
def real: first;
def imag: last;
# Complex or real
def norm:
def sq: .*.;
if type == "number" then sq
else map(sq) | add
end;
# Complex or real
def abs:
if type == "array" then norm | sqrt
elif . < 0 then - . else .
end;
def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;
# Input should be an integer
def isPrime:
. as $n
| if ($n < 2) then false
elif ($n % 2 == 0) then $n == 2
elif ($n % 3 == 0) then $n == 3
else 5
| until( . <= 0;
if .*. > $n then -1
elif ($n % . == 0) then 0
else . + 2
| if ($n % . == 0) then 0
else . + 4
end
end)
| . == -1
end;
# Given a stream of non-null values,
# group by the values of `x|filter` that occur in a run.
def runs(stream; filter):
foreach (stream, null) as $x ({emit: false, array: []};
if $x == null
then .emit = .array
elif .array == [] or ($x|filter) == (.array[0]|filter)
then .array += [$x] | .emit = false
else {emit: .array, array: [$x]}
end;
select(.emit).emit) ;Gaussian Primes
# emit a stream of Gaussian primes with real and imaginary parts within the given radius
def GPrimes($Radius):
def check: abs | isPrime and ((. - 3) % 4 == 0);
($Radius | norm) as $R2
| range(-$Radius; $Radius + 1) as $r
| range(-$Radius; $Radius + 1) as $i
| [$r, $i]
| norm as $norm
| select( $norm < $R2 )
| if $i == 0 then select($r|check)
elif $r == 0 then select($i|check)
else select($norm|isPrime)
end ;
def plot($Radius):
"# X Y",
(GPrimes($Radius) | "\(first) \(last)");Tasks
# For the first task:
"Gaussian primes with norm < 100, grouped by the real part:",
(runs(GPrimes(10); real) | map(tostring) | join(" "))
# For the plot:
# plot(100)- Output:
Invocation: jq -nc -f gaussian-primes.jq
Gaussian primes with norm < 100, grouped by the real part: [-9,-4] [-9,4] [-8,-5] [-8,-3] [-8,3] [-8,5] [-7,-2] [-7,0] [-7,2] [-6,-5] [-6,-1] [-6,1] [-6,5] [-5,-8] [-5,-6] [-5,-4] [-5,-2] [-5,2] [-5,4] [-5,6] [-5,8] [-4,-9] [-4,-5] [-4,-1] [-4,1] [-4,5] [-4,9] [-3,-8] [-3,-2] [-3,0] [-3,2] [-3,8] [-2,-7] [-2,-5] [-2,-3] [-2,-1] [-2,1] [-2,3] [-2,5] [-2,7] [-1,-6] [-1,-4] [-1,-2] [-1,-1] [-1,1] [-1,2] [-1,4] [-1,6] [0,-7] [0,-3] [0,3] [0,7] [1,-6] [1,-4] [1,-2] [1,-1] [1,1] [1,2] [1,4] [1,6] [2,-7] [2,-5] [2,-3] [2,-1] [2,1] [2,3] [2,5] [2,7] [3,-8] [3,-2] [3,0] [3,2] [3,8] [4,-9] [4,-5] [4,-1] [4,1] [4,5] [4,9] [5,-8] [5,-6] [5,-4] [5,-2] [5,2] [5,4] [5,6] [5,8] [6,-5] [6,-1] [6,1] [6,5] [7,-2] [7,0] [7,2] [8,-5] [8,-3] [8,3] [8,5] [9,-4] [9,4]
Julia
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
n = 100;
digs = Reap@Do[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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* * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * * * * * * * *
* * * * * * * * * * ** ** * * * * * * * * * *
* * * * * * * * * * * * * * * *
* * * * * * * * * * ** ** * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
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* * * * * * * * * * * * * * * * * *
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* *
* * * *
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;
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
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
''' 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)
Quackery
eratosthenes and isprime are defined at Sieve of Eratosthenes#Quackery.
[ $ "turtleduck.qky" loadfile ] now!
[ ' [ 0 0 0 ] fill
[ 12 5 circle ] ] is dot ( --> )
[ 6 * 1 fly
-1 4 turn
6 * 1 fly
1 4 turn ] is toxy ( n n --> )
[ 2dup toxy
dot
1 2 turn
toxy
1 2 turn ] is plot ( n n --> )
[ 4 times
[ 300 1 walk
-300 1 fly
1 4 turn ] ] is axes ( --> )
[ dup * ] is ^2 ( n --> n )
[ ^2 swap ^2 + ] is norm ( n n --> n )
4000 eratosthenes
[ 2dup
0 !=
swap 0 != and iff
[ norm isprime ]
done
+ abs
dup 4 mod 3 =
swap isprime and ] is gprime ( n n --> b )
[]
20 ^2 1+ times
[ i^ 20 /mod
10 - dip [ 10 - ]
2dup norm 10 ^2 > iff
2drop done
2dup gprime iff
[ join
nested join ]
else 2drop ]
[] swap
witheach
[ char [ swap
2 times
[ behead rot swap
dup -1 > if
[ swap space join
swap ]
number$ join
swap ]
drop
char ] join
nested join ]
72 wrap$
turtle
0 frames
axes
100 ^2 1+ times
[ i^ 100 /mod
50 - dip [ 50 - ]
2dup norm 50 ^2 > iff
2drop done
2dup gprime iff
plot else 2drop ]
1 frames- Output:
[-9-4] [-9 4] [-8-5] [-8-3] [-8 3] [-8 5] [-7-2] [-7 0] [-7 2] [-6-5] [-6-1] [-6 1] [-6 5] [-5-8] [-5-6] [-5-4] [-5-2] [-5 2] [-5 4] [-5 6] [-5 8] [-4-9] [-4-5] [-4-1] [-4 1] [-4 5] [-4 9] [-3-8] [-3-2] [-3 0] [-3 2] [-3 8] [-2-7] [-2-5] [-2-3] [-2-1] [-2 1] [-2 3] [-2 5] [-2 7] [-1-6] [-1-4] [-1-2] [-1-1] [-1 1] [-1 2] [-1 4] [-1 6] [ 0-7] [ 0-3] [ 0 3] [ 0 7] [ 1-6] [ 1-4] [ 1-2] [ 1-1] [ 1 1] [ 1 2] [ 1 4] [ 1 6] [ 2-7] [ 2-5] [ 2-3] [ 2-1] [ 2 1] [ 2 3] [ 2 5] [ 2 7] [ 3-8] [ 3-2] [ 3 0] [ 3 2] [ 3 8] [ 4-9] [ 4-5] [ 4-1] [ 4 1] [ 4 5] [ 4 9] [ 5-8] [ 5-6] [ 5-4] [ 5-2] [ 5 2] [ 5 4] [ 5 6] [ 5 8] [ 6-5] [ 6-1] [ 6 1] [ 6 5] [ 7-2] [ 7 0] [ 7 2] [ 8-5] [ 8-3] [ 8 3] [ 8 5] [ 9-4] [ 9 4]
Raku
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
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("($ 0.0z) ", 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)