problem 180 // project euler

Rational zeros of a function of three variables

For any integer n, consider the three functions

f_1,n(x,y,z) = xⁿ⁺¹ + yⁿ⁺¹ − zⁿ⁺¹
f_2,n(x,y,z) = (xy + yz + zx)*(x^n-1 + y^n-1 − z^n-1)
f_3,n(x,y,z) = xyz*(x^n-2 + y^n-2 − z^n-2)

and their combination

f_n(x,y,z) = f_1,n(x,y,z) + f_2,n(x,y,z) − f_3,n(x,y,z)

We call (x,y,z) a golden triple of order k if x, y, and z are all rational numbers of the form a / b with
0 < a < b ≤ k and there is (at least) one integer n, so that f_n(x,y,z) = 0.

Let s(x,y,z) = x + y + z.
Let t = u / v be the sum of all distinct s(x,y,z) for all golden triples (x,y,z) of order 35.
All the s(x,y,z) and t must be in reduced form.

Find u + v.

有三个自变量的函数的有理零点

对于任意整数n，考虑这三个函数

f_1,n(x,y,z) = xⁿ⁺¹ + yⁿ⁺¹ − zⁿ⁺¹
f_2,n(x,y,z) = (xy + yz + zx)*(x^n-1 + y^n-1 − z^n-1)
f_3,n(x,y,z) = xyz*(x^n-2 + y^n-2 − z^n-2)

以及它们的组合

f_n(x,y,z) = f_1,n(x,y,z) + f_2,n(x,y,z) − f_3,n(x,y,z)

我们称(x,y,z)为k阶黄金三元组，如果x，y和z都是形如a / b的有理数，其中
0 < a < b ≤ k，且（至少）存在一个整数n使得 f_n(x,y,z) = 0。

令s(x,y,z) = x + y + z。
对于所有不同的35阶黄金三元组(x,y,z)，s(x,y,z)的和记为t = u / v。
所有的s(x,y,z)以及t都是最简分数。

求u + v。