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 $abc$.

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$abc$。