The diophantine equation $a^2+ab-b^2=0$ I first tried with brute force with $-1000 \leq a,b \leq 1000$ but found no solution. 
But then a simple argument showed me that there was no solution. Not only in the integers, but even for the rationals. What was that argument?  

And why is it impossible to solve the equation? $a^2+ab-b^2=0$ 
$$a=\frac{-b\pm\sqrt{b^2+4b^2}}{2}=b\frac{-1\pm\sqrt{5}}{2}$$ 
you can see the whole decision no.
If $a^3+ab-b^2=0$  would have acted similarly. $b^2-ab-a^3=0$
$$b=\frac{a\pm\sqrt{a^2+4a^3}}{2}=a\frac{1\pm\sqrt{4a+1}}{2}=a\frac{1\pm{k}}{2}$$
$a$ - to choose such that the root was intact. $a=\frac{k^2-1}{4}$
If $a^2+pab-b^2=0$
$$a=\frac{-pb\pm\sqrt{p^2b^2+4b^2}}{2}=b\frac{-p\pm\sqrt{p^2+4}}{2}$$
There are solutions when: $p=0 $
