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use strict;
use warnings;
use Const::Fast;
const my $TARGET => 5;
MAIN:
{
my $count = 0;
# It is known that if any odd number n is perfect, n > 10^1500; so only even
# numbers need be considered. By the Euclid-Euler Theorem, an even number n
# is a perfect number if and only if n = 2^(k-1)*(2^k-1), where 2^k-1 is
# prime. So the perfect numbers are a subset of the positive integers n of
# the form n = 2^(k-1)*(2^k-1), where k is a positive integer.
for (my $k = 1; $count < $TARGET; ++$k)
{
my $n = (2 ** ($k - 1)) * (2 ** $k - 1);
if (is_perfect($n))
{
print "$n\n";
++$count;
}
}
}
# A positive integer n is perfect if and only if n is equal to the sum of its
# positive proper divisors (factors). (Equivalently, n is perfect if and only if
# it is equal to half the sum of its divisors, where the latter include n
# itself).
sub is_perfect
{
my ($n) = @_;
return 0 if $n == 1; # 1 is not a perfect number
my $max = int(sqrt($n) + 0.5);
my $sum = 1; # Every positive integer has 1 as a factor
for my $d (2 .. $max)
{
$sum += $d + ($n / $d) if ($n % $d) == 0;
}
return $n == $sum;
}
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