Integer solutions of $$xy = x + y$$ where $$x,y \epsilon Z$$ $$xy - x - y = 0$$ $$(x-1)(y-1) = 1$$ let $$x' = x -1$$ and $$y' = y - 1$$ so $$x',y' \epsilon Z$$ So $$x' = 1 / y'$$ The only integers whose reciprocals are also integers are 1 and -1 So $$y' = 1$$ and $$x' = 1$$ So $$y = 2$$ and $$x = 2$$ AND So $$y' = -1$$ and $$x' = -1$$ So $$y = 0$$ and $$x = 0$$ Therefore there are only two solutions to this diophantine equation $$x = y = 2$$ and $$x = y = 0$$