Cuban primes

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   (3^rd 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   100   cuban primes   (in a multi─line horizontal format).
•   show the   100,000^th   cuban prime.
•   show all cuban primes with commas.
•   show all output here.

Also see

•   MathWorld entry:   cuban prime.
•   The OEIS entry:   A2407.     The   100,000^th   cuban prime can be verified in the   2^nd   example   on this OEIS web page.

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.*/

1. = 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 " " */

1. = 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 x;  leave j;  end /*output is only 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 out0ut*/
    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