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Upside down Diophantine equation

 Published on Saturday, 20th February 2021, 01:00 pm; Solved by 199

Problem 748

Upside Down is a modification of the famous Pythagorean equation: \begin{align} \frac{1}{x^2}+\frac{1}{y^2}=\frac{13}{z^2} \end{align}

A solution $(x,y,z)$ to this equation with $x,y$ and $z$ positive integers is a primitive solution if $\gcd(x,y,z)=1$.

Let $S(N)$ be the sum of $x+y+z$ over primitive Upside Down solutions such that $1 \leq x,y,z \leq N$ and $x \le y$.
For $N=100$ the primitive solutions are $(2,3,6)$ and $(5,90,18)$, thus $S(10^2)=124$.
It can be checked that $S(10^3)=1470$ and $S(10^5)=2340084$.

Find $S(10^{16})$ and give the last 9 digits as your answer.