Cuban primes: Difference between revisions
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Thundergnat (talk | contribs) m (→{{header|Perl 6}}: Remove concurrency and intermediate variables where they don't add anything) |
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for 2..10 -> \k { |
for 2..10 -> \k { |
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next if k %% 3; |
next if k %% 3; |
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my @cubans = lazy (1..Inf |
my @cubans = lazy (1..Inf).map({ (($_+k)³ - .³)/k }).grep: *.is-prime; |
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put "First 20 cuban primes where k = {k}:"; |
put "First 20 cuban primes where k = {k}:"; |
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put @cubans[^20]».&comma».fmt("%7s").rotor(10).join: "\n"; |
put @cubans[^20]».&comma».fmt("%7s").rotor(10).join: "\n"; |
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my \k = 2**128; |
my \k = 2**128; |
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my @cubans = lazy (0..Inf).hyper(:8degree).map({ (($_+k)³ - .³)/k }).grep: *.is-prime; |
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put "First 10 cuban primes where k = {k}:"; |
put "First 10 cuban primes where k = {k}:"; |
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.&comma.put for |
.&comma.put for (lazy (0..Inf).map({ (($_+k)³ - .³)/k }).grep: *.is-prime)[^10];</lang> |
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<pre>First 10 cuban primes where k = 340282366920938463463374607431768211456: |
<pre>First 10 cuban primes where k = 340282366920938463463374607431768211456: |
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115,792,089,237,316,195,423,570,985,008,687,908,160,544,961,995,247,996,546,884,854,518,799,824,856,507 |
115,792,089,237,316,195,423,570,985,008,687,908,160,544,961,995,247,996,546,884,854,518,799,824,856,507 |
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Revision as of 23:06, 5 February 2019
Cuban 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.
The name cuban has nothing to do with Cuba, but has to do with the fact that cubes (3rd powers) play a role in its definition.
- Some definitions of cuban primes
-
- primes which are the difference of two consecutive cubes.
- primes of the form: (n+1)3 - n3.
- primes of the form: n3 - (n-1)3.
- primes p such that n2(p+n) is a cube for some n>0.
- primes p such that 4p = 1 + 3n2.
Cuban primes were named in 1923 by Allan Joseph Champneys Cunningham.
- Task requirements
-
- show the first 200 cuban primes (in a multi─line horizontal format).
- show the 100,000th cuban prime.
- show all cuban primes with commas.
- show all output here.
Note that cuban prime isn't capitalized (as it doesn't refer to Cuba).
- Also see
-
- MathWorld entry: cuban prime.
- The OEIS entry: A2407. The 100,000th cuban prime can be verified in the 2nd example on this OEIS web page.
ALGOL 68
<lang algol68>BEGIN
# find some cuban primes (using the definition: a prime p is a cuban prime if # # p = n^3 - ( n - 1 )^3 # # for some n > 0) #
# returns a string representation of n with commas #
PROC commatise = ( LONG INT n )STRING:
BEGIN
STRING result := "";
STRING unformatted = whole( n, 0 );
INT ch count := 0;
FOR c FROM UPB unformatted BY -1 TO LWB unformatted DO
IF ch count <= 2 THEN ch count +:= 1
ELSE ch count := 1; "," +=: result
FI;
unformatted[ c ] +=: result
OD;
result
END # commatise # ;
# sieve the primes #
INT sieve max = 2 000 000;
[ sieve max ]BOOL sieve; FOR i TO UPB sieve DO sieve[ i ] := TRUE OD;
sieve[ 1 ] := FALSE;
FOR s FROM 2 TO ENTIER sqrt( sieve max ) DO
IF sieve[ s ] THEN
FOR p FROM s * s BY s TO sieve max DO sieve[ p ] := FALSE OD
FI
OD;
# count the primes, we can ignore 2, as we know it isn't a cuban prime #
sieve[ 2 ] := FALSE;
INT prime count := 0;
FOR s TO UPB sieve DO IF sieve[ s ] THEN prime count +:= 1 FI OD;
# construct a list of the primes #
[ 1 : prime count ]INT primes;
INT prime pos := LWB primes;
FOR s FROM LWB sieve TO UPB sieve DO
IF sieve[ s ] THEN primes[ prime pos ] := s; prime pos +:= 1 FI
OD;
# find the cuban primes #
INT cuban count := 0;
LONG INT final cuban := 0;
INT max cuban = 100 000; # mximum number of cubans to find #
INT print limit = 200; # show all cubans up to this one #
print( ( "First ", commatise( print limit ), " cuban primes: ", newline ) );
LONG INT prev cube := 1;
FOR n FROM 2 WHILE
LONG INT this cube = ( LENG n * n ) * n;
LONG INT p = this cube - prev cube;
prev cube := this cube;
IF ODD p THEN
# 2 is not a cuban prime so we only test odd numbers #
BOOL is prime := TRUE;
INT max factor = SHORTEN ENTIER long sqrt( p );
FOR f FROM LWB primes WHILE is prime AND primes[ f ] <= max factor DO
is prime := p MOD primes[ f ] /= 0
OD;
IF is prime THEN
# have a cuban prime #
cuban count +:= 1;
IF cuban count <= print limit THEN
# must show this cuban #
STRING p formatted = commatise( p );
print( ( " "[ UPB p formatted : ], p formatted ) );
IF cuban count MOD 10 = 0 THEN print( ( newline ) ) FI
FI;
final cuban := p
FI
FI;
cuban count < max cuban
DO SKIP OD;
IF cuban count MOD 10 /= 0 THEN print( ( newline ) ) FI;
print( ( "The ", commatise( max cuban ), " cuban prime is: ", commatise( final cuban ), newline ) )
END</lang>
- Output:
First 200 cuban primes:
7 19 37 61 127 271 331 397 547 631
919 1,657 1,801 1,951 2,269 2,437 2,791 3,169 3,571 4,219
4,447 5,167 5,419 6,211 7,057 7,351 8,269 9,241 10,267 11,719
12,097 13,267 13,669 16,651 19,441 19,927 22,447 23,497 24,571 25,117
26,227 27,361 33,391 35,317 42,841 45,757 47,251 49,537 50,311 55,897
59,221 60,919 65,269 70,687 73,477 74,419 75,367 81,181 82,171 87,211
88,237 89,269 92,401 96,661 102,121 103,231 104,347 110,017 112,327 114,661
115,837 126,691 129,169 131,671 135,469 140,617 144,541 145,861 151,201 155,269
163,567 169,219 170,647 176,419 180,811 189,757 200,467 202,021 213,067 231,019
234,361 241,117 246,247 251,431 260,191 263,737 267,307 276,337 279,991 283,669
285,517 292,969 296,731 298,621 310,087 329,677 333,667 337,681 347,821 351,919
360,187 368,551 372,769 374,887 377,011 383,419 387,721 398,581 407,377 423,001
436,627 452,797 459,817 476,407 478,801 493,291 522,919 527,941 553,411 574,219
584,767 590,077 592,741 595,411 603,457 608,851 611,557 619,711 627,919 650,071
658,477 666,937 689,761 692,641 698,419 707,131 733,591 742,519 760,537 769,627
772,669 784,897 791,047 812,761 825,301 837,937 847,477 863,497 879,667 886,177
895,987 909,151 915,769 925,741 929,077 932,419 939,121 952,597 972,991 976,411
986,707 990,151 997,057 1,021,417 1,024,921 1,035,469 1,074,607 1,085,407 1,110,817 1,114,471
1,125,469 1,155,061 1,177,507 1,181,269 1,215,397 1,253,887 1,281,187 1,285,111 1,324,681 1,328,671
1,372,957 1,409,731 1,422,097 1,426,231 1,442,827 1,451,161 1,480,519 1,484,737 1,527,247 1,570,357
The 100,000 cuban prime is: 1,792,617,147,127
C++
<lang Cpp>#include <iostream>
- include <vector>
- include <chrono>
- include <climits>
- include <cmath>
using namespace std;
vector <long long> primes{ 3, 5 };
int main() { cout.imbue(locale("")); const int cutOff = 200, bigUn = 100000, chunks = 50, little = bigUn / chunks;
const char tn[] = " cuban prime";
cout << "The first " << cutOff << tn << "s:" << endl; int c = 0; bool showEach = true; long long u = 0, v = 1; auto st = chrono::system_clock::now();
for (long long i = 1; i <= LLONG_MAX; i++) { bool found = false; long long mx = (long long)(ceil(sqrt(v += (u += 6)))); for (long long item : primes) { if (item > mx) break; if (v % item == 0) { found = true; break; } } if (!found) { c += 1; if (showEach) { for (long long z = primes.back() + 2; z <= v - 2; z += 2) { bool fnd = false; for (long long item : primes) { if (item > mx) break; if (z % item == 0) { fnd = true; break; } } if (!fnd) primes.push_back(z); } primes.push_back(v); cout.width(11); cout << v; if (c % 10 == 0) cout << endl; if (c == cutOff) { showEach = false; cout << "\nProgress to the " << bigUn << "th" << tn << ": "; } } if (c % little == 0) { cout << "."; if (c == bigUn) break; } } } cout << "\nThe " << c << "th" << tn << " is " << v; chrono::duration<double> elapsed_seconds = chrono::system_clock::now() - st; cout << "\nComputation time was " << elapsed_seconds.count() << " seconds" << endl; return 0; }</lang>
- Output:
The first 200 cuban primes:
7 19 37 61 127 271 331 397 547 631
919 1,657 1,801 1,951 2,269 2,437 2,791 3,169 3,571 4,219
4,447 5,167 5,419 6,211 7,057 7,351 8,269 9,241 10,267 11,719
12,097 13,267 13,669 16,651 19,441 19,927 22,447 23,497 24,571 25,117
26,227 27,361 33,391 35,317 42,841 45,757 47,251 49,537 50,311 55,897
59,221 60,919 65,269 70,687 73,477 74,419 75,367 81,181 82,171 87,211
88,237 89,269 92,401 96,661 102,121 103,231 104,347 110,017 112,327 114,661
115,837 126,691 129,169 131,671 135,469 140,617 144,541 145,861 151,201 155,269
163,567 169,219 170,647 176,419 180,811 189,757 200,467 202,021 213,067 231,019
234,361 241,117 246,247 251,431 260,191 263,737 267,307 276,337 279,991 283,669
285,517 292,969 296,731 298,621 310,087 329,677 333,667 337,681 347,821 351,919
360,187 368,551 372,769 374,887 377,011 383,419 387,721 398,581 407,377 423,001
436,627 452,797 459,817 476,407 478,801 493,291 522,919 527,941 553,411 574,219
584,767 590,077 592,741 595,411 603,457 608,851 611,557 619,711 627,919 650,071
658,477 666,937 689,761 692,641 698,419 707,131 733,591 742,519 760,537 769,627
772,669 784,897 791,047 812,761 825,301 837,937 847,477 863,497 879,667 886,177
895,987 909,151 915,769 925,741 929,077 932,419 939,121 952,597 972,991 976,411
986,707 990,151 997,057 1,021,417 1,024,921 1,035,469 1,074,607 1,085,407 1,110,817 1,114,471
1,125,469 1,155,061 1,177,507 1,181,269 1,215,397 1,253,887 1,281,187 1,285,111 1,324,681 1,328,671
1,372,957 1,409,731 1,422,097 1,426,231 1,442,827 1,451,161 1,480,519 1,484,737 1,527,247 1,570,357
Progress to the 100,000th cuban prime: ..................................................
The 100,000th cuban prime is 1,792,617,147,127
Computation time was 35.5644 seconds
C#
(of the Snail Version) <lang csharp>using System; using System.Collections.Generic; using System.Linq;
static class Program {
static List<long> primes = new List<long>() { 3, 5 };
static void Main(string[] args)
{
const int cutOff = 200;
const int bigUn = 100000;
const int chunks = 50;
const int little = bigUn / chunks;
const string tn = " cuban prime";
Console.WriteLine("The first {0:n0}{1}s:", cutOff, tn);
int c = 0;
bool showEach = true;
long u = 0, v = 1;
DateTime st = DateTime.Now;
for (long i = 1; i <= long.MaxValue; i++)
{
bool found = false;
int mx = System.Convert.ToInt32(Math.Ceiling(Math.Sqrt(v += (u += 6))));
foreach (long item in primes)
{
if (item > mx) break;
if (v % item == 0) { found = true; break; }
}
if (!found)
{
c += 1; if (showEach)
{
for (var z = primes.Last() + 2; z <= v - 2; z += 2)
{
bool fnd = false;
foreach (long item in primes)
{
if (item > mx) break;
if (z % item == 0) { fnd = true; break; }
}
if (!fnd) primes.Add(z);
}
primes.Add(v); Console.Write("{0,11:n0}", v);
if (c % 10 == 0) Console.WriteLine();
if (c == cutOff)
{
showEach = false;
Console.Write("\nProgress to the {0:n0}th{1}: ", bigUn, tn);
}
}
if (c % little == 0) { Console.Write("."); if (c == bigUn) break; }
}
}
Console.WriteLine("\nThe {1:n0}th{2} is {0,17:n0}", v, c, tn);
Console.WriteLine("Computation time was {0} seconds", (DateTime.Now - st).TotalSeconds);
if (System.Diagnostics.Debugger.IsAttached) Console.ReadKey();
}
}</lang>
- Output:
The first 200 cuban primes:
7 19 37 61 127 271 331 397 547 631
919 1,657 1,801 1,951 2,269 2,437 2,791 3,169 3,571 4,219
4,447 5,167 5,419 6,211 7,057 7,351 8,269 9,241 10,267 11,719
12,097 13,267 13,669 16,651 19,441 19,927 22,447 23,497 24,571 25,117
26,227 27,361 33,391 35,317 42,841 45,757 47,251 49,537 50,311 55,897
59,221 60,919 65,269 70,687 73,477 74,419 75,367 81,181 82,171 87,211
88,237 89,269 92,401 96,661 102,121 103,231 104,347 110,017 112,327 114,661
115,837 126,691 129,169 131,671 135,469 140,617 144,541 145,861 151,201 155,269
163,567 169,219 170,647 176,419 180,811 189,757 200,467 202,021 213,067 231,019
234,361 241,117 246,247 251,431 260,191 263,737 267,307 276,337 279,991 283,669
285,517 292,969 296,731 298,621 310,087 329,677 333,667 337,681 347,821 351,919
360,187 368,551 372,769 374,887 377,011 383,419 387,721 398,581 407,377 423,001
436,627 452,797 459,817 476,407 478,801 493,291 522,919 527,941 553,411 574,219
584,767 590,077 592,741 595,411 603,457 608,851 611,557 619,711 627,919 650,071
658,477 666,937 689,761 692,641 698,419 707,131 733,591 742,519 760,537 769,627
772,669 784,897 791,047 812,761 825,301 837,937 847,477 863,497 879,667 886,177
895,987 909,151 915,769 925,741 929,077 932,419 939,121 952,597 972,991 976,411
986,707 990,151 997,057 1,021,417 1,024,921 1,035,469 1,074,607 1,085,407 1,110,817 1,114,471
1,125,469 1,155,061 1,177,507 1,181,269 1,215,397 1,253,887 1,281,187 1,285,111 1,324,681 1,328,671
1,372,957 1,409,731 1,422,097 1,426,231 1,442,827 1,451,161 1,480,519 1,484,737 1,527,247 1,570,357
Progress to the 100,000th cuban prime: ..................................................
The 100,000th cuban prime is 1,792,617,147,127
Computation time was 63.578673 seconds
Go
<lang go>package main
import (
"fmt" "math/big"
)
func commatize(n uint64) string {
s := fmt.Sprintf("%d", n)
le := len(s)
for i := le - 3; i >= 1; i -= 3 {
s = s[0:i] + "," + s[i:]
}
return s
}
func main() {
var z big.Int
var cube1, cube2, cube100k, diff uint64
cubans := make([]string, 200)
cube1 = 1
count := 0
for i := 1; ; i++ {
j := i + 1
cube2 = uint64(j * j * j)
diff = cube2 - cube1
z.SetUint64(diff)
if z.ProbablyPrime(0) { // 100% accurate for z < 2 ^ 64
if count < 200 {
cubans[count] = commatize(diff)
}
count++
if count == 100000 {
cube100k = diff
break
}
}
cube1 = cube2
}
fmt.Println("The first 200 cuban primes are:-")
for i := 0; i < 20; i++ {
j := i * 10
fmt.Printf("%9s\n", cubans[j : j+10]) // 10 per line say
}
fmt.Println("\nThe 100,000th cuban prime is", commatize(cube100k))
}</lang>
- Output:
The first 200 cuban primes are:- [ 7 19 37 61 127 271 331 397 547 631] [ 919 1,657 1,801 1,951 2,269 2,437 2,791 3,169 3,571 4,219] [ 4,447 5,167 5,419 6,211 7,057 7,351 8,269 9,241 10,267 11,719] [ 12,097 13,267 13,669 16,651 19,441 19,927 22,447 23,497 24,571 25,117] [ 26,227 27,361 33,391 35,317 42,841 45,757 47,251 49,537 50,311 55,897] [ 59,221 60,919 65,269 70,687 73,477 74,419 75,367 81,181 82,171 87,211] [ 88,237 89,269 92,401 96,661 102,121 103,231 104,347 110,017 112,327 114,661] [ 115,837 126,691 129,169 131,671 135,469 140,617 144,541 145,861 151,201 155,269] [ 163,567 169,219 170,647 176,419 180,811 189,757 200,467 202,021 213,067 231,019] [ 234,361 241,117 246,247 251,431 260,191 263,737 267,307 276,337 279,991 283,669] [ 285,517 292,969 296,731 298,621 310,087 329,677 333,667 337,681 347,821 351,919] [ 360,187 368,551 372,769 374,887 377,011 383,419 387,721 398,581 407,377 423,001] [ 436,627 452,797 459,817 476,407 478,801 493,291 522,919 527,941 553,411 574,219] [ 584,767 590,077 592,741 595,411 603,457 608,851 611,557 619,711 627,919 650,071] [ 658,477 666,937 689,761 692,641 698,419 707,131 733,591 742,519 760,537 769,627] [ 772,669 784,897 791,047 812,761 825,301 837,937 847,477 863,497 879,667 886,177] [ 895,987 909,151 915,769 925,741 929,077 932,419 939,121 952,597 972,991 976,411] [ 986,707 990,151 997,057 1,021,417 1,024,921 1,035,469 1,074,607 1,085,407 1,110,817 1,114,471] [1,125,469 1,155,061 1,177,507 1,181,269 1,215,397 1,253,887 1,281,187 1,285,111 1,324,681 1,328,671] [1,372,957 1,409,731 1,422,097 1,426,231 1,442,827 1,451,161 1,480,519 1,484,737 1,527,247 1,570,357] The 100,000th cuban prime is 1,792,617,147,127
Pascal
uses trial division to check primility.Slow in such number ranges.
OutNthCubPrime(10000) takes only 0,950 s.
100: 283,669; 1000: 65,524,807; 10000: 11,712,188,419; 100000: 1,792,617,147,127
<lang pascal>program CubanPrimes; {$IFDEF FPC}
{$MODE DELPHI}
{$OPTIMIZATION ON,Regvar,PEEPHOLE,CSE,ASMCSE}
{$CODEALIGN proc=32}
{$ENDIF} uses
primTrial;
const
COLUMNCOUNT = 10*10;
procedure FormOut(Cuban:Uint64;ColSize:Uint32); var
s : String; pI,pJ :pChar; i,j : NativeInt;
Begin
str(Cuban,s);
i := length(s);
If i>3 then
Begin
//extend s by the count of comma to be inserted
j := i+ (i-1) div 3;
setlength(s,j);
pI := @s[i];
pJ := @s[j];
while i > 3 do
Begin
// copy 3 digits
pJ^ := pI^;dec(pJ);dec(pI);
pJ^ := pI^;dec(pJ);dec(pI);
pJ^ := pI^;dec(pJ);dec(pI);
// insert comma
pJ^ := ',';dec(pJ);
dec(i,3);
end;
//the digits in front are in the right place
end;
write(s:ColSize);
end;
procedure OutFirstCntCubPrimes(Cnt : Int32;ColCnt : Int32); var
cbDelta1, cbDelta2 : Uint64; ClCnt,ColSize : NativeInt;
Begin
If Cnt <= 0 then EXIT; IF ColCnt <= 0 then ColCnt := 1; ColSize := COLUMNCOUNT DIV ColCnt; dec(ColCnt);
ClCnt := ColCnt; cbDelta1 := 0; cbDelta2 := 1;
repeat
if isPrime(cbDelta2) then
Begin
FormOut(cbDelta2,ColSize);
dec(Cnt);
dec(ClCnt);
If ClCnt < 0 then
Begin
Writeln;
ClCnt := ColCnt;
end;
end;
inc(cbDelta1,6);// 0,6,12,18...
inc(cbDelta2,cbDelta1);//1,7,19,35...
until Cnt<= 0;
writeln;
end;
procedure OutNthCubPrime(n : Int32); var
cbDelta1, cbDelta2 : Uint64;
Begin
If n <= 0 then EXIT; cbDelta1 := 0; cbDelta2 := 1;
repeat
inc(cbDelta1,6);
inc(cbDelta2,cbDelta1);
if isPrime(cbDelta2) then
dec(n);
until n<=0;
FormOut(cbDelta2,20); writeln;
end;
Begin
OutFirstCntCubPrimes(200,10); OutNthCubPrime(100000);
end.</lang>
- Output:
7 19 37 61 127 271 331 397 547 631
919 1,657 1,801 1,951 2,269 2,437 2,791 3,169 3,571 4,219
4,447 5,167 5,419 6,211 7,057 7,351 8,269 9,241 10,267 11,719
12,097 13,267 13,669 16,651 19,441 19,927 22,447 23,497 24,571 25,117
26,227 27,361 33,391 35,317 42,841 45,757 47,251 49,537 50,311 55,897
59,221 60,919 65,269 70,687 73,477 74,419 75,367 81,181 82,171 87,211
88,237 89,269 92,401 96,661 102,121 103,231 104,347 110,017 112,327 114,661
115,837 126,691 129,169 131,671 135,469 140,617 144,541 145,861 151,201 155,269
163,567 169,219 170,647 176,419 180,811 189,757 200,467 202,021 213,067 231,019
234,361 241,117 246,247 251,431 260,191 263,737 267,307 276,337 279,991 283,669
285,517 292,969 296,731 298,621 310,087 329,677 333,667 337,681 347,821 351,919
360,187 368,551 372,769 374,887 377,011 383,419 387,721 398,581 407,377 423,001
436,627 452,797 459,817 476,407 478,801 493,291 522,919 527,941 553,411 574,219
584,767 590,077 592,741 595,411 603,457 608,851 611,557 619,711 627,919 650,071
658,477 666,937 689,761 692,641 698,419 707,131 733,591 742,519 760,537 769,627
772,669 784,897 791,047 812,761 825,301 837,937 847,477 863,497 879,667 886,177
895,987 909,151 915,769 925,741 929,077 932,419 939,121 952,597 972,991 976,411
986,707 990,151 997,057 1,021,417 1,024,921 1,035,469 1,074,607 1,085,407 1,110,817 1,114,471
1,125,469 1,155,061 1,177,507 1,181,269 1,215,397 1,253,887 1,281,187 1,285,111 1,324,681 1,328,671
1,372,957 1,409,731 1,422,097 1,426,231 1,442,827 1,451,161 1,480,519 1,484,737 1,527,247 1,570,357
1,792,617,147,127 //user 2m1.950s
Perl
<lang perl>use feature 'say'; use ntheory 'is_prime';
sub cuban_primes {
my ($n) = @_;
my @primes;
for (my $k = 1 ; ; ++$k) {
my $p = 3 * $k * ($k + 1) + 1;
if (is_prime($p)) {
push @primes, $p;
last if @primes >= $n;
}
}
return @primes;
}
sub commify {
scalar reverse join ',', unpack '(A3)*', reverse shift;
}
my @c = cuban_primes(200);
while (@c) {
say join ' ', map { sprintf "%9s", commify $_ } splice(@c, 0, 10);
}
say ; for my $n (1 .. 6) {
say "10^$n-th cuban prime is: ", commify((cuban_primes(10**$n))[-1]);
}</lang>
- Output:
7 19 37 61 127 271 331 397 547 631
919 1,657 1,801 1,951 2,269 2,437 2,791 3,169 3,571 4,219
4,447 5,167 5,419 6,211 7,057 7,351 8,269 9,241 10,267 11,719
12,097 13,267 13,669 16,651 19,441 19,927 22,447 23,497 24,571 25,117
26,227 27,361 33,391 35,317 42,841 45,757 47,251 49,537 50,311 55,897
59,221 60,919 65,269 70,687 73,477 74,419 75,367 81,181 82,171 87,211
88,237 89,269 92,401 96,661 102,121 103,231 104,347 110,017 112,327 114,661
115,837 126,691 129,169 131,671 135,469 140,617 144,541 145,861 151,201 155,269
163,567 169,219 170,647 176,419 180,811 189,757 200,467 202,021 213,067 231,019
234,361 241,117 246,247 251,431 260,191 263,737 267,307 276,337 279,991 283,669
285,517 292,969 296,731 298,621 310,087 329,677 333,667 337,681 347,821 351,919
360,187 368,551 372,769 374,887 377,011 383,419 387,721 398,581 407,377 423,001
436,627 452,797 459,817 476,407 478,801 493,291 522,919 527,941 553,411 574,219
584,767 590,077 592,741 595,411 603,457 608,851 611,557 619,711 627,919 650,071
658,477 666,937 689,761 692,641 698,419 707,131 733,591 742,519 760,537 769,627
772,669 784,897 791,047 812,761 825,301 837,937 847,477 863,497 879,667 886,177
895,987 909,151 915,769 925,741 929,077 932,419 939,121 952,597 972,991 976,411
986,707 990,151 997,057 1,021,417 1,024,921 1,035,469 1,074,607 1,085,407 1,110,817 1,114,471
1,125,469 1,155,061 1,177,507 1,181,269 1,215,397 1,253,887 1,281,187 1,285,111 1,324,681 1,328,671
1,372,957 1,409,731 1,422,097 1,426,231 1,442,827 1,451,161 1,480,519 1,484,737 1,527,247 1,570,357
10^1-th cuban prime is: 631
10^2-th cuban prime is: 283,669
10^3-th cuban prime is: 65,524,807
10^4-th cuban prime is: 11,712,188,419
10^5-th cuban prime is: 1,792,617,147,127
10^6-th cuban prime is: 255,155,578,239,277
Perl 6
The task (k == 1)
Not the most efficient, but concise, and good enough for this task. <lang perl6>sub comma { $^i.flip.comb(3).join(',').flip }
my @cubans = lazy (1..Inf).hyper(:8degree).map({ ($_+1)³ - .³ }).grep: *.is-prime;
put @cubans[^200]».&comma».fmt("%9s").rotor(10).join: "\n";
put ;
put @cubans[99_999]., # zero indexed</lang>
- Output:
7 19 37 61 127 271 331 397 547 631
919 1,657 1,801 1,951 2,269 2,437 2,791 3,169 3,571 4,219
4,447 5,167 5,419 6,211 7,057 7,351 8,269 9,241 10,267 11,719
12,097 13,267 13,669 16,651 19,441 19,927 22,447 23,497 24,571 25,117
26,227 27,361 33,391 35,317 42,841 45,757 47,251 49,537 50,311 55,897
59,221 60,919 65,269 70,687 73,477 74,419 75,367 81,181 82,171 87,211
88,237 89,269 92,401 96,661 102,121 103,231 104,347 110,017 112,327 114,661
115,837 126,691 129,169 131,671 135,469 140,617 144,541 145,861 151,201 155,269
163,567 169,219 170,647 176,419 180,811 189,757 200,467 202,021 213,067 231,019
234,361 241,117 246,247 251,431 260,191 263,737 267,307 276,337 279,991 283,669
285,517 292,969 296,731 298,621 310,087 329,677 333,667 337,681 347,821 351,919
360,187 368,551 372,769 374,887 377,011 383,419 387,721 398,581 407,377 423,001
436,627 452,797 459,817 476,407 478,801 493,291 522,919 527,941 553,411 574,219
584,767 590,077 592,741 595,411 603,457 608,851 611,557 619,711 627,919 650,071
658,477 666,937 689,761 692,641 698,419 707,131 733,591 742,519 760,537 769,627
772,669 784,897 791,047 812,761 825,301 837,937 847,477 863,497 879,667 886,177
895,987 909,151 915,769 925,741 929,077 932,419 939,121 952,597 972,991 976,411
986,707 990,151 997,057 1,021,417 1,024,921 1,035,469 1,074,607 1,085,407 1,110,817 1,114,471
1,125,469 1,155,061 1,177,507 1,181,269 1,215,397 1,253,887 1,281,187 1,285,111 1,324,681 1,328,671
1,372,957 1,409,731 1,422,097 1,426,231 1,442,827 1,451,161 1,480,519 1,484,737 1,527,247 1,570,357
1,792,617,147,127
k == 2 through 10
After reading up a bit, the general equation for cuban primes is prime numbers of the form ((x+k)3 - x3)/k where k mod 3 is not equal 0.
The cubans where k == 1 (the focus of this task) is one of many possible groups. In general, it seems like the cubans where k == 1 and k == 2 are the two primary cases, but it is possible to have cubans with a k of any integer that is not a multiple of 3.
Here are the first 20 for each valid k up to 10: <lang perl6>sub comma { $^i.flip.comb(3).join(',').flip }
for 2..10 -> \k {
next if k %% 3;
my @cubans = lazy (1..Inf).map({ (($_+k)³ - .³)/k }).grep: *.is-prime;
put "First 20 cuban primes where k = {k}:";
put @cubans[^20]».&comma».fmt("%7s").rotor(10).join: "\n";
put ;
} </lang>
- Output:
First 20 cuban primes where k = 2:
13 109 193 433 769 1,201 1,453 2,029 3,469 3,889
4,801 10,093 12,289 13,873 18,253 20,173 21,169 22,189 28,813 37,633
First 20 cuban primes where k = 4:
31 79 151 367 1,087 1,327 1,879 2,887 3,271 4,111
4,567 6,079 7,207 8,431 15,991 16,879 17,791 19,687 23,767 24,847
First 20 cuban primes where k = 5:
43 67 97 223 277 337 727 823 1,033 1,663
2,113 2,617 2,797 3,373 4,003 5,683 6,217 7,963 10,273 10,627
First 20 cuban primes where k = 7:
73 103 139 181 229 283 409 643 733 829
1,039 1,153 1,399 1,531 1,669 2,281 2,803 3,181 3,583 3,793
First 20 cuban primes where k = 8:
163 379 523 691 883 2,203 2,539 3,691 5,059 5,563
6,091 7,219 8,443 9,091 10,459 11,923 15,139 19,699 24,859 27,091
First 20 cuban primes where k = 10:
457 613 997 1,753 2,053 2,377 4,357 6,373 9,433 13,093
16,453 21,193 27,673 28,837 31,237 37,657 46,153 47,653 49,177 62,233
k == 2^128
Note that Perl 6 has native support for arbitrarily large integers and does not need to generate primes to test for primality. Using k of 2^128; finishes in well under a second. <lang perl6>sub comma { $^i.flip.comb(3).join(',').flip }
my \k = 2**128; put "First 10 cuban primes where k = {k}:"; .&comma.put for (lazy (0..Inf).map({ (($_+k)³ - .³)/k }).grep: *.is-prime)[^10];</lang>
First 10 cuban primes where k = 340282366920938463463374607431768211456: 115,792,089,237,316,195,423,570,985,008,687,908,160,544,961,995,247,996,546,884,854,518,799,824,856,507 115,792,089,237,316,195,423,570,985,008,687,908,174,836,821,405,927,412,012,346,588,030,934,089,763,531 115,792,089,237,316,195,423,570,985,008,687,908,219,754,093,839,491,289,189,512,036,211,927,493,764,691 115,792,089,237,316,195,423,570,985,008,687,908,383,089,629,961,541,751,651,931,847,779,176,235,685,011 115,792,089,237,316,195,423,570,985,008,687,908,491,299,422,642,400,183,033,284,972,942,478,527,291,811 115,792,089,237,316,195,423,570,985,008,687,908,771,011,528,251,411,600,000,178,900,251,391,998,361,371 115,792,089,237,316,195,423,570,985,008,687,908,875,137,932,529,218,769,819,971,530,125,513,071,648,307 115,792,089,237,316,195,423,570,985,008,687,908,956,805,700,590,244,001,051,181,435,909,137,442,897,427 115,792,089,237,316,195,423,570,985,008,687,909,028,264,997,643,641,078,378,490,103,469,808,767,771,907 115,792,089,237,316,195,423,570,985,008,687,909,158,933,426,541,281,448,348,425,952,723,607,761,904,131
REXX
<lang rexx>/*REXX program finds and displays a number of cuban primes or the Nth cuban prime. */ numeric digits 20 /*ensure enought decimal digits for #s.*/ parse arg N . /*obtain optional argument from the CL.*/ if N== | N=="," then N= 200 /*Not specified? Then use the default.*/ Nth= N<0 /*used for finding the Nth cuban prime.*/
- = 0 /*number of cuban primes found so far. */
!.=0; !.0=1; !.2=1; !.3=1; !.4=1; !.5=1; !.6=1; !.8=1; s=41; $= sw= linesize() - 1; if sw<1 then sw=79 /*obtain width of the terminal screen. */ if (Nth & N==1) | N>0 then $= 7 /*handle case of the 1st cuban prime.*/ if (Nth & N==2) | N>1 then $= $ 19 /* " " " " 2nd " " */ if (Nth & N==3) | N>2 then $= $ 37 /* " " " " 3rd " " */
- = 3 /*above are needed for prime generation*/
N= abs(N) /*use absolute value of N from now on*/ if #<N then /* [↑] ending digs that aren't cubans.*/
do j=4 until #=>N; x= (j+1)**3 - j**3 /*compute a possible cuban*/
parse var x -1 _; if !._ then iterate /*check for the last digit*/
if x// 3==0 then iterate; if x// 7==0 then iterate /* " " ÷ 3; for ÷ 7 */
if x//11==0 then iterate; if x//13==0 then iterate /* " " ÷11; " ÷ 13 */
if x//17==0 then iterate; if x//19==0 then iterate /* " " ÷17; " ÷ 19 */
if x//23==0 then iterate; if x//29==0 then iterate /* " " ÷23; " ÷ 29 */
if x//31==0 then iterate; if x//37==0 then iterate /* " " ÷31; " ÷ 37 */
do k=s by 6 until k*k>x /*skip multiples of 3 (when using BY 6)*/
if x// k ==0 then iterate j /*test for a "lower" possible prime. */
if x//(k+2) ==0 then iterate j /* " " " "higher" " " */
end /*k*/
#= # + 1 /*bump the number of cuban primes found*/
if Nth then do; if #==N then do; say commas(x); leave j; end /*display 1 num*/
else iterate j /*keep searching. */
end
new=$ commas(x) /*append a new cuban. */
if length(new)>sw then do; say $ /*line too long, show #s*/
$= commas(x) /*initialize next line. */
end
else $= new /*use this longer output*/
end /*j*/
if S\== then say $ /*check for any residual*/ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ commas: parse arg _; do jc=length(_)-3 to 1 by -3; _=insert(',', _, jc); end; return _</lang>
- output when using the default input of: 200
7 19 37 61 127 271 331 397 547 631 919 1,657 1,801 1,951 2,269 2,437 2,791 3,169 3,571 4,219 4,447 5,167 5,419 6,211 7,057 7,351 8,269 9,241 10,267 11,719 12,097 13,267 13,669 16,651 19,441 19,927 22,447 23,497 24,571 25,117 26,227 27,361 33,391 35,317 42,841 45,757 47,251 49,537 50,311 55,897 59,221 60,919 65,269 70,687 73,477 74,419 75,367 81,181 82,171 87,211 88,237 89,269 92,401 96,661 102,121 103,231 104,347 110,017 112,327 114,661 115,837 126,691 129,169 131,671 135,469 140,617 144,541 145,861 151,201 155,269 163,567 169,219 170,647 176,419 180,811 189,757 200,467 202,021 213,067 231,019 234,361 241,117 246,247 251,431 260,191 263,737 267,307 276,337 279,991 283,669 285,517 292,969 296,731 298,621 310,087 329,677 333,667 337,681 347,821 351,919 360,187 368,551 372,769 374,887 377,011 383,419 387,721 398,581 407,377 423,001 436,627 452,797 459,817 476,407 478,801 493,291 522,919 527,941 553,411 574,219 584,767 590,077 592,741 595,411 603,457 608,851 611,557 619,711 627,919 650,071 658,477 666,937 689,761 692,641 698,419 707,131 733,591 742,519 760,537 769,627 772,669 784,897 791,047 812,761 825,301 837,937 847,477 863,497 879,667 886,177 895,987 909,151 915,769 925,741 929,077 932,419 939,121 952,597 972,991 976,411 986,707 990,151 997,057 1,021,417 1,024,921 1,035,469 1,074,607 1,085,407 1,110,817 1,114,471 1,125,469 1,155,061 1,177,507 1,181,269 1,215,397 1,253,887 1,281,187 1,285,111 1,324,681 1,328,671 1,372,957 1,409,731 1,422,097 1,426,231 1,442,827 1,451,161 1,480,519 1,484,737 1,527,247 1,570,357
- output when using the input of: -100000
1,792,617,147,127
Sidef
<lang ruby>func cuban_primes(n) {
1..Inf -> lazy.map {|k| 3*k*(k+1) + 1 }\
.grep{ .is_prime }\
.first(n)
}
cuban_primes(200).slices(10).each {
say .map { "%9s" % .commify }.join(' ')
}
say ("\n100,000th cuban prime is: ", cuban_primes(1e5).last.commify)</lang>
- Output:
7 19 37 61 127 271 331 397 547 631
919 1,657 1,801 1,951 2,269 2,437 2,791 3,169 3,571 4,219
4,447 5,167 5,419 6,211 7,057 7,351 8,269 9,241 10,267 11,719
12,097 13,267 13,669 16,651 19,441 19,927 22,447 23,497 24,571 25,117
26,227 27,361 33,391 35,317 42,841 45,757 47,251 49,537 50,311 55,897
59,221 60,919 65,269 70,687 73,477 74,419 75,367 81,181 82,171 87,211
88,237 89,269 92,401 96,661 102,121 103,231 104,347 110,017 112,327 114,661
115,837 126,691 129,169 131,671 135,469 140,617 144,541 145,861 151,201 155,269
163,567 169,219 170,647 176,419 180,811 189,757 200,467 202,021 213,067 231,019
234,361 241,117 246,247 251,431 260,191 263,737 267,307 276,337 279,991 283,669
285,517 292,969 296,731 298,621 310,087 329,677 333,667 337,681 347,821 351,919
360,187 368,551 372,769 374,887 377,011 383,419 387,721 398,581 407,377 423,001
436,627 452,797 459,817 476,407 478,801 493,291 522,919 527,941 553,411 574,219
584,767 590,077 592,741 595,411 603,457 608,851 611,557 619,711 627,919 650,071
658,477 666,937 689,761 692,641 698,419 707,131 733,591 742,519 760,537 769,627
772,669 784,897 791,047 812,761 825,301 837,937 847,477 863,497 879,667 886,177
895,987 909,151 915,769 925,741 929,077 932,419 939,121 952,597 972,991 976,411
986,707 990,151 997,057 1,021,417 1,024,921 1,035,469 1,074,607 1,085,407 1,110,817 1,114,471
1,125,469 1,155,061 1,177,507 1,181,269 1,215,397 1,253,887 1,281,187 1,285,111 1,324,681 1,328,671
1,372,957 1,409,731 1,422,097 1,426,231 1,442,827 1,451,161 1,480,519 1,484,737 1,527,247 1,570,357
100,000th cuban prime is: 1,792,617,147,127
Visual Basic .NET
Corner Cutting Version
This language doesn't have a built-in for a IsPrime() function, so I was surprised to find that this runs so quickly. It builds a list of primes while it is creating the output table. Since the last item on the table is larger than the square root of the 100,000th cuban prime, there is no need to continue adding to the prime list while checking up to the 100,000th cuban prime. I found a bit of a shortcut, if you skip the iterator by just the right amount, only one value is tested for the final result. It's hard-coded in the program, so if another final cuban prime were to be selected for output, the program would need a re-write. If not skipping ahead to the answer, it takes a few seconds over a minute to eventually get to it (see Snail Version below). <lang vbnet>Module Module1
Dim primes As List(Of Long) = {3L, 5L}.ToList()
Sub Main(args As String())
Const cutOff As Integer = 200, bigUn As Integer = 100000,
tn As String = " cuban prime"
Console.WriteLine("The first {0:n0}{1}s:", cutOff, tn)
Dim c As Integer = 0, showEach As Boolean = True, skip As Boolean = True,
v As Long = 0, st As DateTime = DateTime.Now
For i As Long = 1 To Long.MaxValue
v = 3 * i : v = v * i + v + 1
Dim found As Boolean = False, mx As Integer = Math.Ceiling(Math.Sqrt(v))
For Each item In primes
If item > mx Then Exit For
If v Mod item = 0 Then found = True : Exit For
Next : If Not found Then
c += 1 : If showEach Then
For z = primes.Last + 2 To v - 2 Step 2
Dim fnd As Boolean = False
For Each item In primes
If item > mx Then Exit For
If z Mod item = 0 Then fnd = True : Exit For
Next : If Not fnd Then primes.Add(z)
Next : primes.Add(v) : Console.Write("{0,11:n0}", v)
If c Mod 10 = 0 Then Console.WriteLine()
If c = cutOff Then showEach = False
Else
If skip Then skip = False : i += 772279 : c = bigUn - 1
End If
If c = bigUn Then Exit For
End If
Next
Console.WriteLine("{1}The {2:n0}th{3} is {0,17:n0}", v, vbLf, c, tn)
Console.WriteLine("Computation time was {0} seconds", (DateTime.Now - st).TotalSeconds)
If System.Diagnostics.Debugger.IsAttached Then Console.ReadKey()
End Sub
End Module</lang>
- Output:
The first 200 cuban primes:
7 19 37 61 127 271 331 397 547 631
919 1,657 1,801 1,951 2,269 2,437 2,791 3,169 3,571 4,219
4,447 5,167 5,419 6,211 7,057 7,351 8,269 9,241 10,267 11,719
12,097 13,267 13,669 16,651 19,441 19,927 22,447 23,497 24,571 25,117
26,227 27,361 33,391 35,317 42,841 45,757 47,251 49,537 50,311 55,897
59,221 60,919 65,269 70,687 73,477 74,419 75,367 81,181 82,171 87,211
88,237 89,269 92,401 96,661 102,121 103,231 104,347 110,017 112,327 114,661
115,837 126,691 129,169 131,671 135,469 140,617 144,541 145,861 151,201 155,269
163,567 169,219 170,647 176,419 180,811 189,757 200,467 202,021 213,067 231,019
234,361 241,117 246,247 251,431 260,191 263,737 267,307 276,337 279,991 283,669
285,517 292,969 296,731 298,621 310,087 329,677 333,667 337,681 347,821 351,919
360,187 368,551 372,769 374,887 377,011 383,419 387,721 398,581 407,377 423,001
436,627 452,797 459,817 476,407 478,801 493,291 522,919 527,941 553,411 574,219
584,767 590,077 592,741 595,411 603,457 608,851 611,557 619,711 627,919 650,071
658,477 666,937 689,761 692,641 698,419 707,131 733,591 742,519 760,537 769,627
772,669 784,897 791,047 812,761 825,301 837,937 847,477 863,497 879,667 886,177
895,987 909,151 915,769 925,741 929,077 932,419 939,121 952,597 972,991 976,411
986,707 990,151 997,057 1,021,417 1,024,921 1,035,469 1,074,607 1,085,407 1,110,817 1,114,471
1,125,469 1,155,061 1,177,507 1,181,269 1,215,397 1,253,887 1,281,187 1,285,111 1,324,681 1,328,671
1,372,957 1,409,731 1,422,097 1,426,231 1,442,827 1,451,161 1,480,519 1,484,737 1,527,247 1,570,357
The 100,000th cuban prime is 1,792,617,147,127
Computation time was 0.2989494 seconds
Snail Version
This one doesn't take any shortcuts. It could be sped up (Execution time about 15 seconds) by threading chunks of the search for the 100,000th cuban prime, but you would have to take a guess about how far to go, which would be hard-coded, so one might as well use the short-cut version if you are willing to overlook that difficulty. <lang vbnet>Module Program
Dim primes As List(Of Long) = {3L, 5L}.ToList()
Sub Main(args As String())
Dim taskList As New List(Of Task(Of Integer))
Const cutOff As Integer = 200, bigUn As Integer = 100000,
chunks As Integer = 50, little As Integer = bigUn / chunks,
tn As String = " cuban prime"
Console.WriteLine("The first {0:n0}{1}s:", cutOff, tn)
Dim c As Integer = 0, showEach As Boolean = True,
u As Long = 0, v As Long = 1,
st As DateTime = DateTime.Now
For i As Long = 1 To Long.MaxValue
u += 6 : v += u
Dim found As Boolean = False, mx As Integer = Math.Ceiling(Math.Sqrt(v))
For Each item In primes
If item > mx Then Exit For
If v Mod item = 0 Then found = True : Exit For
Next : If Not found Then
c += 1 : If showEach Then
For z = primes.Last + 2 To v - 2 Step 2
Dim fnd As Boolean = False
For Each item In primes
If item > mx Then Exit For
If z Mod item = 0 Then fnd = True : Exit For
Next : If Not fnd Then primes.Add(z)
Next : primes.Add(v) : Console.Write("{0,11:n0}", v)
If c Mod 10 = 0 Then Console.WriteLine()
If c = cutOff Then showEach = False : _
Console.Write("{0}Progress to the {1:n0}th{2}: ", vbLf, bigUn, tn)
End If
If c Mod little = 0 Then Console.Write(".") : If c = bigUn Then Exit For
End If
Next
Console.WriteLine("{1}The {2:n0}th{3} is {0,17:n0}", v, vbLf, c, tn)
Console.WriteLine("Computation time was {0} seconds", (DateTime.Now - st).TotalSeconds)
If System.Diagnostics.Debugger.IsAttached Then Console.ReadKey()
End Sub
End Module</lang>
- Output:
The first 200 cuban primes:
7 19 37 61 127 271 331 397 547 631
919 1,657 1,801 1,951 2,269 2,437 2,791 3,169 3,571 4,219
4,447 5,167 5,419 6,211 7,057 7,351 8,269 9,241 10,267 11,719
12,097 13,267 13,669 16,651 19,441 19,927 22,447 23,497 24,571 25,117
26,227 27,361 33,391 35,317 42,841 45,757 47,251 49,537 50,311 55,897
59,221 60,919 65,269 70,687 73,477 74,419 75,367 81,181 82,171 87,211
88,237 89,269 92,401 96,661 102,121 103,231 104,347 110,017 112,327 114,661
115,837 126,691 129,169 131,671 135,469 140,617 144,541 145,861 151,201 155,269
163,567 169,219 170,647 176,419 180,811 189,757 200,467 202,021 213,067 231,019
234,361 241,117 246,247 251,431 260,191 263,737 267,307 276,337 279,991 283,669
285,517 292,969 296,731 298,621 310,087 329,677 333,667 337,681 347,821 351,919
360,187 368,551 372,769 374,887 377,011 383,419 387,721 398,581 407,377 423,001
436,627 452,797 459,817 476,407 478,801 493,291 522,919 527,941 553,411 574,219
584,767 590,077 592,741 595,411 603,457 608,851 611,557 619,711 627,919 650,071
658,477 666,937 689,761 692,641 698,419 707,131 733,591 742,519 760,537 769,627
772,669 784,897 791,047 812,761 825,301 837,937 847,477 863,497 879,667 886,177
895,987 909,151 915,769 925,741 929,077 932,419 939,121 952,597 972,991 976,411
986,707 990,151 997,057 1,021,417 1,024,921 1,035,469 1,074,607 1,085,407 1,110,817 1,114,471
1,125,469 1,155,061 1,177,507 1,181,269 1,215,397 1,253,887 1,281,187 1,285,111 1,324,681 1,328,671
1,372,957 1,409,731 1,422,097 1,426,231 1,442,827 1,451,161 1,480,519 1,484,737 1,527,247 1,570,357
Progress to the 100,000th cuban prime: ..................................................
The 100,000th cuban prime is 1,792,617,147,127
Computation time was 49.5868152 seconds
k > 1 Version
A VB.NET version of the Perl 6 version where k > 1, linked at Try It Online!