my $Limit = 1000000;
my $HighestFactor = sqrt($Limit);
my $is_prime='';                  # Sieve array.
# put in candidate primes: integers which have an odd number of repres
+entations by certain quadratic forms.
for $x ( 1..$HighestFactor){
    my $x2 = $x*$x;
    last if $x2*2 >= $Limit;
    for $y ( 1..$HighestFactor ){
      my $y2 = $y*$y;
      next if ($n = 3*$x2 - $y2) > $Limit;
      vec($is_prime,$n,1) ^= 1 if $x > $y &&  $n % 12 == 11;
      next if ( $n = 3*$x2 + $y2 ) > $Limit;
      vec($is_prime,$n,1) ^= 1 if $n % 12 == 7;
      next if ( $n = 4*$x2 + $y2 ) > $Limit;
      vec($is_prime,$n,1) ^= 1 if $n % 12 == 1 || $n % 12 == 5;
  }
}
# eliminate composites by sieving
# if n is prime, omit all multiples of its square; this is sufficient 
+because
# composites which managed to get on the list cannot be square-free
for $n (5..$HighestFactor ){
    next unless vec($is_prime,$n,1);
    for( $k=$n2=$n*$n; $k <= $Limit; $k += $n2 ){ vec($is_prime,$k,1) 
+= 0 };
}
# Present the results.
$\="\n";
print for 2,3, grep vec($is_prime,$_,1), 5..$Limit;
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