positive integer (x,y)

[man111]

the number of positive integer solution $(x,y)$ of the equation $\displaystyle \frac{1}{x^2}+\frac{1}{xy}+\frac{1}{y^2} = 1$

[ewert]

0.

[Dave]

$$(x=1) \lor (y=1) \Rightarrow \frac{1}{x^2}+\frac{1}{xy}+\frac{1}{y^2} > 1,$$

$$(x \ge 2) \land (y \ge 2) \Rightarrow \frac{1}{x^2}+\frac{1}{xy}+\frac{1}{y^2} \le \frac 3 4.$$

[man111]

thanks ewert and Dave