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How many integer solutions ( in terms of integer $a,b,c$) has the equation $$ x \cdot y \cdot z =(a-x) (b-y) (c-z), $$ here $x,y,z>0$ and $a-x,b-y,c-y>0$.

I can find number of solutions for some values $a,b,c$ but I hope there exists a formula or a generating function.

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  • $\begingroup$ $a=2x,b=2y,c=2z$ works, but there must be others. What solutions have you found? $\endgroup$ – Old Peter Apr 16 '17 at 15:20
  • $\begingroup$ No, $a,b,c$ are fixed constants, I am looking for $x,y,z$. $\endgroup$ – Leox Apr 16 '17 at 15:37
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For the solution of the equation.

$$xyz=(a-x)(b-y)(c-z)$$

Solution we write expanding on the multipliers. $c=qt$ And $sn=\frac{q-1}{2}$ ; $pk=\frac{q+1}{2}$

$$x=n(bp-as)$$

$$y=s(ak-bn)$$

$$z=tpk$$

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  • $\begingroup$ Do we get all solutions under this parametrisation? $\endgroup$ – Leox Apr 17 '17 at 18:15

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