prog.pl

#!/usr/bin/perl

# a(n) is the least k such that k^j+2 is prime for j = 1 to n but not n+1.
# https://oeis.org/A359396

# Known terms:
#   5, 9, 105, 3, 909, 4995825, 28212939

# a(8) > 10^11. - Lucas A. Brown, Jan 11 2023

use 5.014;
use strict;
use warnings;

use Math::GMPz;
use ntheory qw(:all);

my $n = 8;
my $z = Math::GMPz::Rmpz_init();

forprimes {

    if (($_-2) % 3 == 0) {

        my $count = 1;
        my $j     = 2;
        my $m     = $_-2;

        while (1) {
            Math::GMPz::Rmpz_ui_pow_ui($z, $m, $j);
            Math::GMPz::Rmpz_add_ui($z, $z, 2);
            Math::GMPz::Rmpz_probab_prime_p($z, 0) || last;
            ++$j;
            ++$count;
        }

        if ($count >= $n) {
            die "a($count) = $m";
        }
    }

} 1e11, 1e13;