Problem 9

Special Pythagorean Triplet

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 $a b c$ .

毕达哥拉斯三元组由三个自然数 $a < b < c$ 组成，并满足

$a^{2} + b^{2} = c^{2}$

例如， $3^{2} + 4^{2} = 9 + 16 = 25 = 5^{2}$ 。

有且只有一个毕达哥拉斯三元组满足 $a + b + c = 1000$ 。求这个三元组的乘积 $a b c$ 。