Eisenstein primes: Difference between revisions
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ω is generally chosen as a cube root of unity: <math>{\displaystyle \omega ={\frac {-1+i{\sqrt {3}}}{2}}=e^{i2\pi /3}}</math>
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).
Line 12:
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 <em>norm</em>:
norm(eisenstein integer(a, b)) = |a + bω|² = <math>a^2 - ab + b^2</math>
;Task
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=={{header|J}}==
Implementation:
<syntaxhighlight lang=J>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
}}</syntaxhighlight>
Task example (and stretch - taking the stretch goal in a minimalist literal fashion):
<syntaxhighlight lang=J> 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</syntaxhighlight>
[[File:J-2000-eisenstein-primes.png]]
=={{header|jq}}==
'''Works with jq, the C implementation of jq'''
'''Works with gojq, the Go implementation of jq'''
'''Adapted from [[#Wren|Wren]]'''
In this entry, complex numbers are represented as arrays of pairs: [real, complex],
as in the [[Arithmetic/Complex#jq | 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:
<pre>
reset
set terminal pngcairo
set output "eisenstein-primes.png"
</pre>
<syntaxhighlight lang="jq">
### 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
</syntaxhighlight>
{{output}}
The results of `listing` are shown below.
For the graph of the output produced by `graph`, see the graph shown above at [[#J|J]].
<pre>
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
</pre>
=={{header|Julia}}==
Line 76 ⟶ 279:
lim = isqrt(graphlimitsquared)
arr = [Eisenstein(a, b) for a = -lim:lim, b = -lim:lim]
eprimes = sort!(
for (i, c) in enumerate(eprimes)
if i <= printlimit
Line 118 ⟶ 320:
</pre>
[[File:Plot eisenstein primes julia example.svg|center|Eisenstein primes]]
=={{header|Nim}}==
{{trans|Wren}}
{{libheader|gnuplotlib.nim}}
<syntaxhighlight lang="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")
</syntaxhighlight>
{{out}}
<pre>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
</pre>
[[File:Eisenstein primes (Nim).png|center|Eisenstein primes]]
=={{header|Perl}}==
{{trans|Raku}}
<syntaxhighlight lang="perl" line>
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];</syntaxhighlight>
{{out}}
<pre>
-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
</pre>
=={{header|Phix}}==
{{libheader|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.
<!--<syntaxhighlight lang="phix">(phixonline)-->
<span style="color: #7060A8;">requires</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"1.0.3"</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">include</span> <span style="color: #004080;">complex</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span>
<span style="color: #008080;">constant</span> <span style="color: #000000;">OMEGA</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{-</span><span style="color: #000000;">0.5</span><span style="color: #0000FF;">,</span> <span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">3</span><span style="color: #0000FF;">)*</span><span style="color: #000000;">0.5</span><span style="color: #0000FF;">}</span>
<span style="color: #000080;font-style:italic;">// try to replicate Wren sort order for easy comparison</span>
<span style="color: #008080;">enum</span> <span style="color: #000000;">NORM</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">IMAG</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">REAL</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">IS_PRIME</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">ELEN</span><span style="color: #0000FF;">=$</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">new_Eisenstein</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">real</span><span style="color: #0000FF;">,</span><span style="color: #000000;">imag</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">complex_add</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">complex_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">OMEGA</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">),</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">norm</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">*</span><span style="color: #000000;">a</span><span style="color: #0000FF;">-</span><span style="color: #000000;">a</span><span style="color: #0000FF;">*</span><span style="color: #000000;">b</span><span style="color: #0000FF;">+</span><span style="color: #000000;">b</span><span style="color: #0000FF;">*</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">max</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">abs</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">),</span><span style="color: #7060A8;">abs</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">))</span>
<span style="color: #004080;">bool</span> <span style="color: #000000;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">or</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">or</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">=</span><span style="color: #000000;">b</span><span style="color: #0000FF;">?</span><span style="color: #7060A8;">is_prime</span><span style="color: #0000FF;">(</span><span style="color: #000000;">c</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">and</span> <span style="color: #7060A8;">remainder</span><span style="color: #0000FF;">(</span><span style="color: #000000;">c</span><span style="color: #0000FF;">,</span><span style="color: #000000;">3</span><span style="color: #0000FF;">)=</span><span style="color: #000000;">2</span>
<span style="color: #0000FF;">:</span><span style="color: #7060A8;">is_prime</span><span style="color: #0000FF;">(</span><span style="color: #000000;">norm</span><span style="color: #0000FF;">))</span>
<span style="color: #008080;">return</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">norm</span><span style="color: #0000FF;">,</span><span style="color: #000000;">imag</span><span style="color: #0000FF;">,</span><span style="color: #000000;">real</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">}</span> <span style="color: #000080;font-style:italic;">-- nb in [NORM..IS_PRIME] order</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">Eisenstein</span><span style="color: #0000FF;">()</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">eprimes</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{}</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">=-</span><span style="color: #000000;">100</span> <span style="color: #008080;">to</span> <span style="color: #000000;">100</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">=-</span><span style="color: #000000;">100</span> <span style="color: #008080;">to</span> <span style="color: #000000;">100</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">e</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">new_Eisenstein</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">IS_PRIME</span><span style="color: #0000FF;">]</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">eprimes</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">append</span><span style="color: #0000FF;">(</span><span style="color: #000000;">eprimes</span><span style="color: #0000FF;">,</span><span style="color: #000000;">e</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #000000;">eprimes</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sort</span><span style="color: #0000FF;">(</span><span style="color: #000000;">eprimes</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">real</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">vslice</span><span style="color: #0000FF;">(</span><span style="color: #000000;">eprimes</span><span style="color: #0000FF;">,</span><span style="color: #000000;">REAL</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">imag</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">vslice</span><span style="color: #0000FF;">(</span><span style="color: #000000;">eprimes</span><span style="color: #0000FF;">,</span><span style="color: #000000;">IMAG</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">f100</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #008000;">""</span><span style="color: #0000FF;">,</span><span style="color: #000000;">100</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">100</span> <span style="color: #008080;">do</span> <span style="color: #000080;font-style:italic;">-- convert for display</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">pm</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">imag</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]>=</span><span style="color: #000000;">0</span><span style="color: #0000FF;">?</span><span style="color: #008000;">'+'</span><span style="color: #0000FF;">:</span><span style="color: #008000;">'-'</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">f100</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"%7.4f %c%7.4fi"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">real</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">pm</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">abs</span><span style="color: #0000FF;">(</span><span style="color: #000000;">imag</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">])})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">return</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">f100</span><span style="color: #0000FF;">,</span><span style="color: #000000;">real</span><span style="color: #0000FF;">,</span><span style="color: #000000;">imag</span><span style="color: #0000FF;">}</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #004080;">sequence</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">f100</span><span style="color: #0000FF;">,</span><span style="color: #000000;">real</span><span style="color: #0000FF;">,</span><span style="color: #000000;">imag</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">Eisenstein</span><span style="color: #0000FF;">()</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"First 100 Eisenstein primes nearest zero:\n%s\n"</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">join_by</span><span style="color: #0000FF;">(</span><span style="color: #000000;">f100</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">4</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" "</span><span style="color: #0000FF;">))</span>
<span style="color: #008080;">include</span> <span style="color: #000000;">xpGUI</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">get_data</span><span style="color: #0000FF;">(</span><span style="color: #004080;">gdx</span> <span style="color: #000000;">graph</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">return</span> <span style="color: #0000FF;">{{</span><span style="color: #000000;">real</span><span style="color: #0000FF;">,</span><span style="color: #000000;">imag</span><span style="color: #0000FF;">}}</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #004080;">gdx</span> <span style="color: #000000;">graph</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">gGraph</span><span style="color: #0000FF;">(</span><span style="color: #000000;">get_data</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"XMIN=-150,XMAX=150,YMIN=-100,YMAX=100"</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">dlg</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">gDialog</span><span style="color: #0000FF;">(</span><span style="color: #000000;">graph</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Eisenstein primes"</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"SIZE=392x290"</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">gSetAttribute</span><span style="color: #0000FF;">(</span><span style="color: #000000;">graph</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"GTITLE"</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"with norm <= 100 (%d points)"</span><span style="color: #0000FF;">,{</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">real</span><span style="color: #0000FF;">)})</span>
<span style="color: #7060A8;">gSetAttributes</span><span style="color: #0000FF;">(</span><span style="color: #000000;">graph</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"XTICK=50,YTICK=25,MARKSTYLE=DOT,GRID=NO"</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">gShow</span><span style="color: #0000FF;">(</span><span style="color: #000000;">dlg</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">gMainLoop</span><span style="color: #0000FF;">()</span>
<!--</syntaxhighlight>-->
Output same as Wren.
=={{header|Raku}}==
<syntaxhighlight lang="raku" line>
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;
</syntaxhighlight>
{{out}}
<pre>
-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
</pre>
=={{header|Sidef}}==
{{trans|Raku}}
<syntaxhighlight lang="ruby" line>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(' ')
}</syntaxhighlight>
{{out}}
<pre>
-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
</pre>
=={{header|Wren}}==
{{libheader|DOME}}
{{libheader|Wren-plot}}
{{libheader|Wren-iterate}}
{{libheader|Wren-complex}}
{{libheader|Wren-math}}
{{libheader|Wren-fmt}}
<syntaxhighlight lang="wren">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()</syntaxhighlight>
{{out}}
Terminal output:
<pre>
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
</pre>
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