<s>[INST] Find all integers $x$, $y$, and $z$ such that $\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$ Characterize all positive integers $x$, $y$, and $z$ such that:
$$\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{z}$$
For example, $\dfrac{1}{x+1} + \dfrac{1}{x(x+1)} =  \dfrac{1}{x}$.
 [/INST] For the equation.
$$\frac{1}{x}+\frac{1}{y}=\frac{1}{z}$$
The easiest way is to factor the square.
$$p^2=ks$$
Then decisions can be recorded.
$$x=p$$
$$y=s-p$$
$$z=p-k$$
For the more General case formula there.  Number of solution for $xy +yz + zx = N$ 
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