[euler/prob009] tighten loop constraint and add explanation

Author: gerdr

euler/prob009-gerdr.pl

@@ -1,16 +1,42 @@
 use v6;
 
-sub first-triple(\N) {
-	for 1..(N div 3) -> \a {
+# Let (a, b, c) be a pythagorean triple
+#
+#	a < b < c
+#	a² + b² = c²
+#
+# For N = a + b + c it follows
+#
+#	b = N·(N - 2a) / 2·(N - a)
+#	c = N·(N - 2a) / 2·(N - a) + a²/(N - a)
+#
+# which automatically meets b < c.
+#
+# The condition a < b gives the constraint
+#
+#	a < (1 - 1/√2)·N
+#
+# which we use in the form
+#
+#	a < (2·N - √(2·N²)) / 2
+#
+# to minimize computational errors.
+
+sub triples(\N) {
+	my \A = Int(2 * N - sqrt(2 * N * N)) div 2;
+
+	for 1..A -> \a {
 		my \u = N * (N - 2 * a);
 		my \v = 2 * (N - a);
 
+		# check if b = u/v is an integer
+		# if so, we've found a triple
 		if u %% v {
 			my \b = u div v;
 			my \c = N - a - b;
-			return a, b, c;
+			take $(a, b, c);
 		}
 	}
 }
 
-say [*] first-triple(1000);
+say [*] .list for gather triples(1000);