For the equation: $$\frac{4}{q}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$ The solution can be written using the factorization, as follows. $$p^2-s^2=(p-s)(p+s)=2qL$$ Then the solutions have the form: $$x=\frac{p(p-s)}{4L-q}$$ $$y=\frac{p(p+s)}{4L-q}$$ $$z=L$$ I usually choose the number $ L $ such that the difference: $ (4L-q) $ was equal to: $ 1,2,3,4 $ Although your desire you can choose other. You can write a little differently. If unfold like this: $$p^2-s^2=(p-s)(p+s)=qL$$ The solutions have the form: $$x=\frac{2p(p-s)}{4L-q}$$ $$y=\frac{2p(p+s)}{4L-q}$$ $$z=L$$


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