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

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SUMMARY

The equation $\frac{1}{x^2} + \frac{1}{xy} + \frac{1}{y^2} = 1$ has been proven to have no solutions for natural numbers $x$ and $y$. The discussion highlights the impossibility of satisfying this equation under the constraints of $x, y \in \mathbb{N}$. Key mathematical principles, including properties of fractions and natural number behavior, were utilized to arrive at this conclusion.

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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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