user
The name   **cuban**   has nothing to do with   Cuba  (the country),   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   4*p* = 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   (if appropriate).
        * show all output here.

Note that   **cuban prime**   isn't capitalized   (as it doesn't refer to the nation of Cuba).

Also see
:   * Wikipedia entry:     cuban prime.
    * 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.
assistant
(formerly Perl 6)

**Library:** ntheory

### 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.

```
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].&comma; # zero indexed

```

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:

```
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 '';
}

```

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 Raku 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.

```
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];

```

```
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
```