Problem Statement

In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form Fn = 22n + 1 where n is a non-negative integer. Despite the simplicity of generating Fermat numbers, they have some powerful mathematical properties and are extensively used in cryptography & pseudo-random number generation, and are often linked to other number theoric fields. As of this writing, (mid 2019), there are only five known prime Fermat numbers, the first five (F0 through F4). Only the first twelve Fermat numbers have been completely factored, though many have been partially factored.

Let's start with the solution:

use strict;
use warnings;
use feature 'say';
use bigint try=>"GMP";
use ntheory qw<factor>;

my @Fermats = map { 2**(2**$_) + 1 } 0..9;

my $sub = 0;
say 'First 10 Fermat numbers:';
printf "F%s = %s\n", $sub++, $_ for @Fermats;

$sub = 0;
say "\nFactors of first few Fermat numbers:";
for my $f (map { [factor($_)] } @Fermats[0..8]) {
   printf "Factors of F%s: %s\n", $sub++, @$f == 1 ? 'prime' : join ' ', @$f
}