Project Euler 9

题目

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, \(32 + 42 = 9 + 16 = 25 = 52\).

There exists exactly one Pythagorean triplet for which \(a + b + c = 1000\).

Find the product \(abc\).

解决方案

设\(n=1000\)，并联立以下两个方程： \[\left \{ \begin{aligned} a^2 + b^2 &= c^2\\ a+b+c &=n \end{aligned} \right. \] 可以得到\(b\)关于\(a\)的关系： \[b=\dfrac{n^2-2an}{2(n-a)}\]

因此，由\(a\)可以直接计算出\(b\)。通过判断\(b\)是否为整数，然后计算出\(c\)，再判断是否满足\(a<b<c\)即可。

代码

N = 1000
ans = 0
for a in range(1, N // 3):
    if (N * N - 2 * a * N) % (2 * N - 2 * a) == 0:
        b = (N * N - 2 * a * N) // (2 * N - 2 * a)
        c = N - a - b
        if a < b < c:
            ans += a * b * c
print(ans)