Perfect numbers: Difference between revisions

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 In addition to generating even perfect numbers, we can also have a fast function which returns true when a given even number is perfect:
-<lang perl>use ntheory qw(is_mersenne_prime valuation hammingweight is_power sqrtint);
+<lang perl>use ntheory qw(is_mersenne_prime valuation);
 sub is_even_perfect {
     my ($n) = @_;
+    my $v = valuation($n, 2) || return;
-    $n % 2 == 0 || return;
+    my $m = ($n >> $v);
+    ($m & ($m + 1)) && return;
-    my $square = 8 * $n + 1;
+    ($m >> $v) == 1 || return;
+    is_mersenne_prime($v + 1);
-    is_power($square, 2) || return;
-    my $k = (sqrtint($square) + 1) / 2;
-    hammingweight($k) == 1 && is_mersenne_prime(valuation($k, 2));
 }</lang>