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 squarefree
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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