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. – PROJECT EULER #9