# Special Pythagorean triplet

*Author: Gerhard R*

https://projecteuler.net/problem=9

A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,

a^2 + b^2 = c^2

For example, 3^2 + 4^2 = 9 + 16 = 25 = 5^2.

There exists exactly one Pythagorean triplet for which a + b + c = 1000. Find the product abc.

Solution method:

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

Source code: prob009-gerdr.pl

use v6; sub triples($N) { for 1..Int((1 - sqrt(0.5)) * $N) -> $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; take $($a, $b, $c); } } } say [*] .list for gather triples(1000);