<s>[INST] Show $x^2 +xy-y^2 = 0$ is only true when $x$ & $y$ are zero. Show that it is impossible to find non-zero integers $x$ and $y$ satisfying $x^2 +xy-y^2 = 0$.
 [/INST] Let $g=\gcd(x,y)$.  Let $x=rg$ and $y=sg$.  Then
$$g^2r^2+g^2rs-g^2s^2=0$$
$$r^2+rs-s^2=0$$
$$s^2-r^2=rs$$
$$(s+r)(s-r)=rs$$
We've divided out all common factors, so $\gcd(r,s)=1$.  For any prime $p$, if $p|r$, $p$ does not divide $s$ and therefore divides neither $s+r$ nor $s-r$.  The same argument applies if $p|s$.  Therefore, $r$ and $s$ contain no prime factors and $r=s=1.$  Since $r=s$, $x=y$. So
$$x^2+xy-y^2=x^2+x^2-x^2=x^2=0$$
$$x=0$$
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