Pierpont primes

Pierpont 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.

A Pierpont prime is a prime number of the form: 2^u3^v + 1 for some non-negative integers u and v .

A Pierpont prime of the second kind is a prime number of the form: 2^u3^v - 1 for some non-negative integers u and v .

The term "Pierpont primes" is generally understood to mean the first definition, but will be called "Pierpont primes of the first kind" on this page to distinguish them.

Task

• Write a routine (function, procedure, whatever) to find Pierpont primes of the first & second kinds.

• Use the routine to find and display here, on this page, the first 50 Pierpont primes of the first kind.

• Use the routine to find and display here, on this page, the first 50 Pierpont primes of the second kind

• If your language supports large integers, find and show here, on this page, the 250^th Pierpont prime of the first kind and the 250^th Pierpont prime of the second kind.

See also

• Wikipedia - Pierpont primes
• OEIS:A005109 - Class 1 -, or Pierpont primes
• OEIS:A005105 - Class 1 +, or Pierpont primes of the second kind

Perl 6

Works with: Rakudo version 2019.07.1

<lang perl6>use ntheory:from<Perl5> <is_prime>;

sub pierpont ($type is copy = 1) {

   gather {
       fail "Unknown type: $type Must be one of 1 (default) or 2" if $type !== 1|2;
       take 2 if $type == 1;
       $type = -1 if $type == 2;
       state $po3 = 0;
       state $add-one = 3;
       state @iterators = [2,4,8 … *].iterator, [3,9,27 … *].iterator;

       my @head = @iterators».pull-one;

       loop {
           my $key = @head[*]»[0].pairs.min( *.value ).key;
           my $min = @head[$key];
           @head[$key] = @iterators[$key].pull-one;

           take $min + $type if "{$min + $type}".&is_prime;

           if $min >= $add-one {
                ++$po3;
                @iterators.push: ([|((2,4,8).map: * * 3 ** $po3) … *]).iterator;
                @head[+@iterators - 1] = @iterators[+@iterators - 1].pull-one;
                $add-one *= 3;
           }
       }
   }

}

say "First 50 Pierpont primes of the first kind:\n" ~ pierpont[^50].rotor(10)».fmt('%8d').join: "\n";

say "\nFirst 50 Pierpont primes of the second kind:\n" ~ pierpont(2)[^50].rotor(10)».fmt('%8d').join: "\n";

say "\n250th Pierpont prime of the first kind: " ~ pierpont[249];

say "\n250th Pierpont prime of the second kind: " ~ pierpont(2)[249];</lang>

Output:

First 50 Pierpont primes of the first kind:
       2        3        5        7       13       17       19       37       73       97
     109      163      193      257      433      487      577      769     1153     1297
    1459     2593     2917     3457     3889    10369    12289    17497    18433    39367
   52489    65537   139969   147457   209953   331777   472393   629857   746497   786433
  839809   995329  1179649  1492993  1769473  1990657  2654209  5038849  5308417  8503057

First 50 Pierpont primes of the second kind:
       2        3        5        7       11       17       23       31       47       53
      71      107      127      191      383      431      647      863      971     1151
    2591     4373     6143     6911     8191     8747    13121    15551    23327    27647
   62207    73727   131071   139967   165887   294911   314927   442367   472391   497663
  524287   786431   995327  1062881  2519423 10616831 17915903 18874367 25509167 30233087

250th Pierpont prime of the first kind: 62518864539857068333550694039553

250th Pierpont prime of the second kind: 4111131172000956525894875083702271

REXX

The REXX language has a "big num" capability to handle almost any amount of decimal digits,   but
it lacks a robust   isPrime   function.   Without that, verifying very large primes is problematic. <lang rexx>/*REXX program finds and displays Pierpont primes of the first and second kinds. */ parse arg n . /*obtain optional argument from the CL.*/ if n== | n=="," then n= 50 /*Not specified? Then use the default.*/ numeric digits n /*ensure enough decimal digs (bit int).*/ big= copies(9, digits() ) /*BIG: used as a max number (a limit).*/

      do t=1  to -1  by -2;  usum= 0;   vsum= 0;    s= 0
      #= 0                                      /*number of Pierpont primes  (so far). */
      w= 0                                      /*the max width of a Pierpont prime.   */
      $=;    do j=0  until #>=n                 /*$: the list of the Pierpont primes.  */
             if usum<=s  then usum= get(2, 3);    if vsum<=s  then vsum= get(3, 2)
             s= min(vsum, usum);  if \isPrime(s)  then iterate  /*get min;  is prime?  */
             #= # + 1;            $= $ s                        /*bump counter; append.*/
             w= max(w, length(s) )                              /*find max prime width.*/
             end   /*j*/
      say
      if t==1  then @= '1st'                                    /*choose word for type.*/
               else @= '2nd'                                    /*   "     "   "    "  */
      say center(n   " Pierpont primes of the "   @   ' kind', max(10*(w+1) -1, 79), "═")
      call show $                                               /*display the primes.  */
      end   /*type*/

exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ show: do j=1 by 10 to words($); _=

          do k=j  for 10;                 _= _ right( word($, k), w)
          end   /*k*/
        if _\==  then say substr( strip(_, 'T'), 2)
        end     /*j*/;                                        return

/*──────────────────────────────────────────────────────────────────────────────────────*/ isPrime: procedure; parse arg x; if x<2 then return 0 /*not a prime.*/

        if wordpos(x, '2 3 5 7')\==0     then return 1                  /*it's a prime.*/
        if x//2==0  then return 0;       if x//3==0  then return 0      /*not  a prime.*/
          do j=5  by 6  until j*j>x
          if x//j==0  then return 0;     if x//(j+2)==0  then return 0  /*not  a prime.*/
          end   /*j*/;                                        return 1  /*it's a prime.*/

/*──────────────────────────────────────────────────────────────────────────────────────*/ get: parse arg c1,c2; m=big; do ju=0; pu= c1**ju; if pu+t>s then return min(m, pu+t)

                               do jv=0;  pv= c2**jv;  if pv  >s  then iterate ju
                               _= pu*pv + t;          if _   >s  then m= min(_, m)
                               end   /*jv*/
                             end     /*ju*/           /*see the    RETURN    (above).  */</lang>

output   when using the default input:

═════════════════════50  Pierpont primes of the  1st  kind═════════════════════
      2       3       5       7      13      17      19      37      73      97
    109     163     193     257     433     487     577     769    1153    1297
   1459    2593    2917    3457    3889   10369   12289   17497   18433   39367
  52489   65537  139969  147457  209953  331777  472393  629857  746497  786433
 839809  995329 1179649 1492993 1769473 1990657 2654209 5038849 5308417 8503057

══════════════════════════50  Pierpont primes of the  2nd  kind══════════════════════════
       2        3        5        7       11       17       23       31       47       53
      71      107      127      191      383      431      647      863      971     1151
    2591     4373     6143     6911     8191     8747    13121    15551    23327    27647
   62207    73727   131071   139967   165887   294911   314927   442367   472391   497663
  524287   786431   995327  1062881  2519423 10616831 17915903 18874367 25509167 30233087