<s>[INST] 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?  
 [/INST] 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 $
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