<s>[INST] $x^2+xy+y^2$ and $x^2-xy+y^2$ are not both perfect squares 
Prove that $x^2+xy+y^2$ and $x^2-xy+y^2$ cannot be both perfect squares.

Surely $x$ and $y$ are natural numbers. If $x^2+xy+y^2 =a^2$ and $x^2-xy+y^2=b^2$ simultaneously then we have to show that there are no such integers $a$ and $b$.
I have tried that:
Suppose $x^2+xy+y^2=a^2$ and $x^2-xy+y^2=b^2$. Then $2xy=a^2-b^2=(a+b)(a-b)$, so one of $x$ and $y$ is even, but then I am stuck.
 [/INST] Adding these equations gives $2(x^2+y^2) = a^2 + b^2$; unique factoriation of Gaussian integers shows that we essentially must have $a = x+y$ and $b = x-y$. Now combine this with what you already have.
</s>