MHB Prove 1/x^2+1/xy+1/y^2=1 has no real solution

[Albert1]

$x,y\in N,\,\, and \,\,\dfrac {1}{x^2}+\dfrac{1}{xy}+\dfrac {1}{y^2}=1----(1)$

prove $(1)$ has no solution

 

[kaliprasad]

$x,y\in N,\,\, and \,\,\dfrac {1}{x^2}+\dfrac{1}{xy}+\dfrac {1}{y^2}=1----(1)$

prove $(1)$ has no solution

Spoiler

neither x nor y can be 1 as LHS >= 1.

so x , y >= 2 and for x = 2 , y =2 LHS = $\frac{3}{4}$ so LHS < 1 so cannot be 1 higeer x ,y lower is the value