Problem 785

Symmetric Diophantine equation

Consider the following Diophantine equation:
$$15 (x^2 + y^2 + z^2) = 34 (xy + yz + zx)$$
where $x$, $y$ and $z$ are positive integers.

Let $S(N)$ be the sum of all solutions, $(x,y,z)$, of this equation such that, $1 \le x \le y \le z \le N$ and $\gcd(x, y, z) = 1$.

For $N = 10^2$, there are three such solutions - $(1, 7, 16), (8, 9, 39), (11, 21, 72)$. So $S(10^2) = 184$.

Find $S(10^9)$.

考虑如下丢番图方程：
$$15 (x^2 + y^2 + z^2) = 34 (xy + yz + zx)$$
其中$x$、$y$、$z$均为正整数。

对于上述方程满足$1 \le x \le y \le z \le N$和$\gcd(x, y, z) = 1$的解$(x,y,z)$，记$S(N)$为所有这些解之和。

对于$N = 10^2$，共有三组这样的解：$(1, 7, 16), (8, 9, 39), (11, 21, 72)$，因此$S(10^2) = 184$。

求$S(10^9)$。