<s>[INST] Finding all positive integer solutions for $x+y=xyz-1$ How do I manually solve $x+y=xyz-1$ assuming that $x, y$ and $z$ are positive integers? I was able to guess all possible solutions, but I do not know how to show that these are the only ones:
$x=1, y=1, z=3$
$x=1, y=2, z=2$
$x=2, y=1, z=2$
$x=2, y=3, z=1$
$x=3, y=2, z=1$  
Any hints would be appreciated.
 [/INST] Note that for $x,y$  positive integers $$xy\ge x+y+1$$ for $x>2$ and $y\ge 3$ (or vice verse). Therefore you can control only $y=1$(or vice verse) and $y=x=2$
If $x=1$ the equation becomes:
$$y(z-1)=1$$ and the solutions are $z=2$ $y=2$ and $z=3$, $y=1$. The same thing for $y=1$.  While for $y=x=2$ there aren't solutions.
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