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

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The equation 1/x^2 + 1/xy + 1/y^2 = 1 is analyzed for natural number solutions x and y. By manipulating the equation, it can be shown that the left-hand side is always less than 1 for positive integers. The terms decrease as x and y increase, indicating that the sum cannot equal 1. Therefore, there are no natural number solutions for the equation. The conclusion is that the equation has no real solutions in the specified domain.
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$x,y\in N,\,\, and \,\,\dfrac {1}{x^2}+\dfrac{1}{xy}+\dfrac {1}{y^2}=1----(1)$

prove $(1)$ has no solution
 
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Albert said:
$x,y\in N,\,\, and \,\,\dfrac {1}{x^2}+\dfrac{1}{xy}+\dfrac {1}{y^2}=1----(1)$

prove $(1)$ has no solution

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
 

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