Cuban primes: Difference between revisions
| Line 1,167: | Line 1,167: | ||
printf(1,"\nThe %,dth cuban prime is %s\n",{np,mpz_get_str(pn,comma_fill:=true)}) |
printf(1,"\nThe %,dth cuban prime is %s\n",{np,mpz_get_str(pn,comma_fill:=true)}) |
||
randstate = gmp_randclear(randstate) |
randstate = gmp_randclear(randstate) |
||
{p3,i3,p} = |
{p3,i3,p} = mpz_free({p3,i3,p}) |
||
?elapsed(time()-t0)</lang> |
?elapsed(time()-t0)</lang> |
||
{{out}} |
{{out}} |
||
Revision as of 21:10, 1 August 2019
You are encouraged to solve this task according to the task description, using any language you may know.
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 the nation of Cuba).
- Also see
-
- MathWorld entry: cuban prime.
- The OEIS entry: A002407. 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 c>#include <limits.h>
- include <math.h>
- include <stdbool.h>
- include <stdio.h>
- include <stdlib.h>
typedef long long llong_t; struct PrimeArray {
llong_t *ptr; size_t size; size_t capacity;
};
struct PrimeArray allocate() {
struct PrimeArray primes;
primes.size = 0; primes.capacity = 10; primes.ptr = malloc(primes.capacity * sizeof(llong_t));
return primes;
}
void deallocate(struct PrimeArray *primes) {
free(primes->ptr); primes->ptr = NULL;
}
void push_back(struct PrimeArray *primes, llong_t p) {
if (primes->size >= primes->capacity) {
size_t new_capacity = (3 * primes->capacity) / 2 + 1;
llong_t *temp = realloc(primes->ptr, new_capacity * sizeof(llong_t));
if (NULL == temp) {
fprintf(stderr, "Failed to reallocate the prime array.");
exit(1);
} else {
primes->ptr = temp;
primes->capacity = new_capacity;
}
}
primes->ptr[primes->size++] = p;
}
int main() {
const int cutOff = 200, bigUn = 100000, chunks = 50, little = bigUn / chunks; struct PrimeArray primes = allocate(); int c = 0; bool showEach = true; llong_t u = 0, v = 1, i;
push_back(&primes, 3); push_back(&primes, 5);
printf("The first %d cuban primes:\n", cutOff);
for (i = 1; i < LLONG_MAX; ++i) {
bool found = false;
llong_t mx = ceil(sqrt(v += (u += 6)));
llong_t j;
for (j = 0; j < primes.size; ++j) {
if (primes.ptr[j] > mx) {
break;
}
if (v % primes.ptr[j] == 0) {
found = true;
break;
}
}
if (!found) {
c += 1;
if (showEach) {
llong_t z;
for (z = primes.ptr[primes.size - 1] + 2; z <= v - 2; z += 2) {
bool fnd = false;
for (j = 0; j < primes.size; ++j) {
if (primes.ptr[j] > mx) {
break;
}
if (z % primes.ptr[j] == 0) {
fnd = true;
break;
}
}
if (!fnd) {
push_back(&primes, z);
}
}
push_back(&primes, v);
printf("%11lld", v);
if (c % 10 == 0) {
printf("\n");
}
if (c == cutOff) {
showEach = false;
printf("\nProgress to the %dth cuban prime: ", bigUn);
}
}
if (c % little == 0) {
printf(".");
if (c == bigUn) {
break;
}
}
}
}
printf("\nThe %dth cuban prime is %lld\n", c, v);
deallocate(&primes); return 0;
}</lang>
- Output:
The first 200 cuban primes:
7 19 37 61 127 271 331 397 547 631
919 1657 1801 1951 2269 2437 2791 3169 3571 4219
4447 5167 5419 6211 7057 7351 8269 9241 10267 11719
12097 13267 13669 16651 19441 19927 22447 23497 24571 25117
26227 27361 33391 35317 42841 45757 47251 49537 50311 55897
59221 60919 65269 70687 73477 74419 75367 81181 82171 87211
88237 89269 92401 96661 102121 103231 104347 110017 112327 114661
115837 126691 129169 131671 135469 140617 144541 145861 151201 155269
163567 169219 170647 176419 180811 189757 200467 202021 213067 231019
234361 241117 246247 251431 260191 263737 267307 276337 279991 283669
285517 292969 296731 298621 310087 329677 333667 337681 347821 351919
360187 368551 372769 374887 377011 383419 387721 398581 407377 423001
436627 452797 459817 476407 478801 493291 522919 527941 553411 574219
584767 590077 592741 595411 603457 608851 611557 619711 627919 650071
658477 666937 689761 692641 698419 707131 733591 742519 760537 769627
772669 784897 791047 812761 825301 837937 847477 863497 879667 886177
895987 909151 915769 925741 929077 932419 939121 952597 972991 976411
986707 990151 997057 1021417 1024921 1035469 1074607 1085407 1110817 1114471
1125469 1155061 1177507 1181269 1215397 1253887 1281187 1285111 1324681 1328671
1372957 1409731 1422097 1426231 1442827 1451161 1480519 1484737 1527247 1570357
Progress to the 100000th cuban prime: ..................................................
The 100000th cuban prime is 1792617147127
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
D
<lang d>import std.math; import std.stdio;
void main() {
long[] primes = [3, 5];
immutable cutOff = 200;
immutable bigUn = 100_000;
immutable chunks = 50;
immutable little = bigUn / chunks;
immutable tn = " cuban prime";
writefln("The first %s%ss:", cutOff, tn);
int c;
bool showEach = true;
long u;
long v = 1;
for (long i = 1; i > 0; ++i) {
bool found;
u += 6;
v += u;
int mx = cast(int)ceil(sqrt(cast(real)v));
foreach (item; primes) {
if (item > mx) break;
if (v % item == 0) {
found = true;
break;
}
}
if (!found) {
c++;
if (showEach) {
for (auto z = primes[$-1] + 2; z <= v - 2; z += 2) {
bool fnd;
foreach (item; primes) {
if (item > mx) break;
if (z % item == 0) {
fnd = true;
break;
}
}
if (!fnd) {
primes ~= z;
}
}
primes ~= v;
writef("%11d", v);
if (c % 10 == 0) writeln;
if (c == cutOff) {
showEach = false;
writef("\nProgress to the %sth%s: ", bigUn, tn);
}
}
if (c % little == 0) {
write('.');
if (c == bigUn) {
break;
}
}
}
}
writefln("\nThe %sth%s is %17s", c, tn, v);
}</lang>
- Output:
The first 200 cuban primes:
7 19 37 61 127 271 331 397 547 631
919 1657 1801 1951 2269 2437 2791 3169 3571 4219
4447 5167 5419 6211 7057 7351 8269 9241 10267 11719
12097 13267 13669 16651 19441 19927 22447 23497 24571 25117
26227 27361 33391 35317 42841 45757 47251 49537 50311 55897
59221 60919 65269 70687 73477 74419 75367 81181 82171 87211
88237 89269 92401 96661 102121 103231 104347 110017 112327 114661
115837 126691 129169 131671 135469 140617 144541 145861 151201 155269
163567 169219 170647 176419 180811 189757 200467 202021 213067 231019
234361 241117 246247 251431 260191 263737 267307 276337 279991 283669
285517 292969 296731 298621 310087 329677 333667 337681 347821 351919
360187 368551 372769 374887 377011 383419 387721 398581 407377 423001
436627 452797 459817 476407 478801 493291 522919 527941 553411 574219
584767 590077 592741 595411 603457 608851 611557 619711 627919 650071
658477 666937 689761 692641 698419 707131 733591 742519 760537 769627
772669 784897 791047 812761 825301 837937 847477 863497 879667 886177
895987 909151 915769 925741 929077 932419 939121 952597 972991 976411
986707 990151 997057 1021417 1024921 1035469 1074607 1085407 1110817 1114471
1125469 1155061 1177507 1181269 1215397 1253887 1281187 1285111 1324681 1328671
1372957 1409731 1422097 1426231 1442827 1451161 1480519 1484737 1527247 1570357
Progress to the 100000th cuban prime: ..................................................
The 100000th cuban prime is 1792617147127
F#
The functions
This task uses Extensible Prime Generator (F#) <lang fsharp> // Generate cuban primes. Nigel Galloway: June 9th., 2019 let cubans=Seq.unfold(fun n->Some(n*n*n,n+1L)) 1L|>Seq.pairwise|>Seq.map(fun(n,g)->g-n)|>Seq.filter(isPrime64) let cL=let g=System.Globalization.CultureInfo("en-GB") in (fun (n:int64)->n.ToString("N0",g)) </lang>
The Task
<lang fsharp> cubans|>Seq.take 200|>List.ofSeq|>List.iteri(fun n g->if n%8=7 then printfn "%12s" (cL(g)) else printf "%12s" (cL(g))) </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
<lang fsharp> printfn "\n\n%s" (cL(Seq.item 99999 cubans)) </lang>
- Output:
1,792,617,147,127
Factor
<lang factor>USING: formatting grouping io kernel lists lists.lazy math math.primes sequences tools.memory.private ; IN: rosetta-code.cuban-primes
- cuban-primes ( n -- seq )
1 lfrom [ [ 3 * ] [ 1 + * ] bi 1 + ] <lazy-map> [ prime? ] <lazy-filter> ltake list>array ;
200 cuban-primes 10 <groups> [ [ commas ] map [ "%10s" printf ] each nl ] each nl
1e5 cuban-primes last commas "100,000th cuban prime is: %s\n" printf</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
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
J
I've used assertions to demonstrate and to prove the defined verbs <lang j>
isPrime =: 1&p: assert 1 0 -: isPrime 3 9
NB. difference, but first cube, of incremented y with y
dcc =: -&(^&3)~ >:
assert ((8 9 13^3)-7 8 12^3) -: dcc 7 8 12
Filter =: (#~`)(`:6) assert 2 3 5 7 11 13 -: isPrime Filter i. 16
cubanPrime =: [: isPrime Filter dcc assert 7 19 37 61 127 271 331 397 547 631 919 -: cubanPrime i. 20
NB. comatose copies with comma fill comatose =: (#!.','~ (1 1 1j1 1 1 1j1 1 1 1j1 1 1 1j1 1 1 1j1 1 1 1j1 1 1 1 {.~ -@:#))@:":&> assert (comatose 1000 1238 12 989832) -: [;._2 ] 0 :0 1,000 1,238 12 989,832 )
CP =: cubanPrime i. 800000x # CP NB. tally, I've stored more than 100000 cuban primes
103278
NB. granted, I used wolframalpha Solve[(n+1)^3-n^3==1792617147127,n]
9!:17]2 2 NB. specify bottom right position in box
comatose&.> 10 20 $ CP
┌─────────┬─────────┬─────────┬─────────┬─────────┬─────────┬─────────┬─────────┬─────────┬─────────┬─────────┬─────────┬─────────┬─────────┬─────────┬─────────┬─────────┬─────────┬─────────┬─────────┐ │ 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│ └─────────┴─────────┴─────────┴─────────┴─────────┴─────────┴─────────┴─────────┴─────────┴─────────┴─────────┴─────────┴─────────┴─────────┴─────────┴─────────┴─────────┴─────────┴─────────┴─────────┘
NB. the one hundred thousandth cuban prime
comatose (<: 100000) { CP
1,792,617,147,127
cubanPrime f. NB. cubanPrime with fixed adverbs
[: (#~ 1&p:) (-&(^&3)~ >:)
</lang>
Julia
<lang julia>using Formatting, Primes
function cubanprimes(N)
cubans = zeros(BigInt, N)
cube100k = 0
cube1 = BigInt(1)
count = 1
for i in 1:100000000
j = BigInt(i + 1)
cube2 = j^3
diff = cube2 - cube1
if isprime(diff)
if count <= N
cubans[count] = diff
end
if count == 100000
cube100k = diff
break
end
count += 1
end
cube1 = cube2
end
println("The first $N cuban primes are: ")
foreach(x -> print(lpad(cubans[x] == 0 ? "" : cubans[x], 10), x % 8 == 0 ? "\n" : ""), 1:N)
println("\nThe 100,000th cuban prime is ", format(cube100k, commas=true))
end
cubanprimes(200)
</lang>
- Output:
The first 200 cuban primes are:
7 19 37 61 127 271 331 397
547 631 919 1657 1801 1951 2269 2437
2791 3169 3571 4219 4447 5167 5419 6211
7057 7351 8269 9241 10267 11719 12097 13267
13669 16651 19441 19927 22447 23497 24571 25117
26227 27361 33391 35317 42841 45757 47251 49537
50311 55897 59221 60919 65269 70687 73477 74419
75367 81181 82171 87211 88237 89269 92401 96661
102121 103231 104347 110017 112327 114661 115837 126691
129169 131671 135469 140617 144541 145861 151201 155269
163567 169219 170647 176419 180811 189757 200467 202021
213067 231019 234361 241117 246247 251431 260191 263737
267307 276337 279991 283669 285517 292969 296731 298621
310087 329677 333667 337681 347821 351919 360187 368551
372769 374887 377011 383419 387721 398581 407377 423001
436627 452797 459817 476407 478801 493291 522919 527941
553411 574219 584767 590077 592741 595411 603457 608851
611557 619711 627919 650071 658477 666937 689761 692641
698419 707131 733591 742519 760537 769627 772669 784897
791047 812761 825301 837937 847477 863497 879667 886177
895987 909151 915769 925741 929077 932419 939121 952597
972991 976411 986707 990151 997057 1021417 1024921 1035469
1074607 1085407 1110817 1114471 1125469 1155061 1177507 1181269
1215397 1253887 1281187 1285111 1324681 1328671 1372957 1409731
1422097 1426231 1442827 1451161 1480519 1484737 1527247 1570357
The 100,000th cuban prime is 1,792,617,147,127
Pascal
uses trial division to check primility.Slow in such number ranges.
OutNthCubPrime(10000) takes only 0,950 s.
100: 283,669; 1000: 65,524,807; 10000: 11,712,188,419; 100000: 1,792,617,147,127
<lang pascal>program CubanPrimes; {$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. Use the ntheory library for prime testing; gets it down to around 20 seconds. <lang perl6>use Lingua::EN::Numbers; use ntheory:from<Perl5> <:all>;
my @cubans = lazy (1..Inf).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
Phix
<lang Phix>include mpfr.e integer np = 0,
i = 2
mpz p3 = mpz_init(1*1*1),
i3 = mpz_init(), p = mpz_init(), pn = mpz_init()
atom randstate = gmp_randinit_mt()
printf(1,"The first 200 cuban primes are:\n") sequence first200 = {} atom t0 = time() while np<100000 do
mpz_ui_pow_ui(i3,i,3)
mpz_sub(p,i3,p3)
if mpz_probable_prime_p(p,randstate) then
mpz_set(pn,p)
np += 1
if np<=200 then
first200 = append(first200,sprintf("%,9d",mpz_get_integer(pn)))
if mod(np,10)=0 then
printf(1,"%s\n",join(first200[-10..-1]))
end if
end if
end if
mpz_set(p3,i3)
i += 1
end while printf(1,"\nThe %,dth cuban prime is %s\n",{np,mpz_get_str(pn,comma_fill:=true)}) randstate = gmp_randclear(randstate) {p3,i3,p} = mpz_free({p3,i3,p}) ?elapsed(time()-t0)</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
"5.9s"
Python
<lang Python> import datetime import math
primes = [ 3, 5 ]
cutOff = 200
bigUn = 100_000 chunks = 50 little = bigUn / chunks
tn = " cuban prime" print ("The first {:,}{}s:".format(cutOff, tn))
c = 0 showEach = True u = 0 v = 1 st = datetime.datetime.now()
for i in range(1, int(math.pow(2,20))): found = False u += 6 v += u mx = int(math.sqrt(v))
for item in primes: if (item > mx): break if (v % item == 0): found = True break
if (found == 0): c += 1 if (showEach): z = primes[-1] while (z <= v - 2): z += 2
fnd = False for item in primes: if (item > mx): break if (z % item == 0): fnd = True break
if (not fnd): primes.append(z)
primes.append(v) print("{:>11,}".format(v), end=)
if (c % 10 == 0): print(""); if (c == cutOff): showEach = False print ("Progress to the {:,}th {}:".format(bigUn, tn), end=) if (c % little == 0): print('.', end=) if (c == bigUn): break
print(""); print ("The {:,}th{} is {:,}".format(c, tn, v)) print("Computation time was {} seconds".format((datetime.datetime.now() - st).seconds)) </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 856 seconds
REXX
<lang rexx>/*REXX program finds and displays a number of cuban primes or the Nth cuban prime. */ numeric digits 20 /*ensure enough 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
Ruby
<lang ruby>require "openssl"
RE = /(\d)(?=(\d\d\d)+(?!\d))/ # Activesupport uses this for commatizing cuban_primes = Enumerator.new do |y|
(1..).each do |n| cand = 3*n*(n+1) + 1 y << cand if OpenSSL::BN.new(cand).prime? end
end
def commatize(num)
num.to_s.gsub(RE, "\\1,")
end
cbs = cuban_primes.take(200) formatted = cbs.map{|cb| commatize(cb).rjust(10) } puts formatted.each_slice(10).map(&:join)
t0 = Time.now puts " 100_000th cuban prime is #{commatize( cuban_primes.take(100_000).last)} which took #{(Time.now-t0).round} seconds to calculate."</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
which took 31 seconds to calculate.
Scala
In this example, we start by building an infinite lazy list of cubans and filter out non-primes. This gives us a lazily evaluated list of all cuban primes, and finding the first 200 simply involves taking 200 elements off the list.
To find the 100,000th cuban prime, performance becomes an issue. To remedy this, we write a function that breaks off a chunk from the front of the list of cubans and filters it using a parallel vector, repeating this process until it's found enough cuban primes. This allows us to benefit from the memory efficiency of lazy lists and the number-crunching speed of parallel vectors at the same time.
Spire's SafeLong is used instead of Java's BigInt for performance. <lang scala>import spire.math.SafeLong import spire.implicits._
import scala.annotation.tailrec import scala.collection.parallel.immutable.ParVector
object CubanPrimes {
def main(args: Array[String]): Unit = {
println(formatTable(cubanPrimes.take(200).toVector, 10))
println(f"The 100,000th cuban prime is: ${getNthCubanPrime(100000).toBigInt}%,d")
}
def cubanPrimes: LazyList[SafeLong] = cubans.filter(isPrime)
def cubans: LazyList[SafeLong] = LazyList.iterate(SafeLong(0))(_ + 1).map(n => (n + 1).pow(3) - n.pow(3))
def isPrime(num: SafeLong): Boolean = (num > 1) && !(SafeLong(2) #:: LazyList.iterate(SafeLong(3)){n => n + 2}).takeWhile(n => n*n <= num).exists(num%_ == 0)
def getNthCubanPrime(num: Int): SafeLong = {
@tailrec
def nHelper(rem: Int, src: LazyList[SafeLong]): SafeLong = {
val cprimes = src.take(100000).to(ParVector).filter(isPrime)
if(cprimes.size < rem) nHelper(rem - cprimes.size, src.drop(100000))
else cprimes.toVector.sortWith(_<_)(rem - 1)
}
nHelper(num, cubans)
}
def formatTable(lst: Vector[SafeLong], rlen: Int): String = {
@tailrec
def fHelper(ac: Vector[String], src: Vector[String]): String = {
if(src.nonEmpty) fHelper(ac :+ src.take(rlen).mkString, src.drop(rlen))
else ac.mkString("\n")
}
val maxLen = lst.map(n => f"${n.toBigInt}%,d".length).max
val formatted = lst.map(n => s"%,${maxLen + 2}d".format(n.toInt))
fHelper(Vector[String](), formatted)
}
}</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
The 100,000th cuban prime is: 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!
zkl
Using GMP (GNU Multiple Precision Arithmetic Library, probabilistic primes), because it is easy and fast to test for primeness.
Extensible prime generator#zkl could be used instead. <lang zkl>var [const] BI=Import("zklBigNum"); // libGMP cubans:=(1).walker(*).tweak('wrap(n){ // lazy iterator
p:=3*n*(n + 1) + 1; BI(p).probablyPrime() and p or Void.Skip
}); println("First 200 cuban primes:"); do(20){ (10).pump(String, cubans.next, "%10,d".fmt).println() }
cubans.drop(100_000 - cubans.n).value :
println("\nThe 100,000th cuban prime is: %,d".fmt(_));</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 100,000th cuban prime is: 1,792,617,147,127
Now lets get big. <lang zkl>k,z := BI(2).pow(128), 10; println("First %d cuban primes where k = %,d:".fmt(z,k)); foreach n in ([BI(1)..]){
p:=( (k + n).pow(3) - n.pow(3) )/k;
if(p.probablyPrime()){ println("%,d".fmt(p)); z-=1; }
if(z<=0) break;
}</lang>
- Output:
First 10 cuban primes where k = 340,282,366,920,938,463,463,374,607,431,768,211,456: 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