009 - Special Pythagorean Triplet

Problem 9 / Challenge #9: Special Pythagorean Triplet

A Pythagorean triplet is a set of three natural numbers, $a \lt b \lt c$, 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 $abc$.

ProjectEuler+ Problem Statement

The Project Euler problem is equivalent to the ProjectEuler+ challenge with $T = 1$ and $N = 1000$.