<s>[INST] Prove that there are infinitely many perfect cubes of the form $p^2+3q^2$ Prove that there are infinitely many perfect cubes of the form $p^2+3q^2$ where $p$ and $q$ are integers.
Hint: one approach is to set $p^2+3q^2=(a^2+3b^2)^3$ and then find $(p,q)$ in terms of $a,b$.
Any different approach is very welcome!
 [/INST] Use this from cubing Eusenstein integers, of the form $a+b\sqrt{\;-3}$.
For $a, b$ as integers, then 
$$(a+b\sqrt{-3})^3 = (a^3 - 9ab^2) + (3a^2b - 3b^3)\sqrt{-3} $$
Whereapon, one sees that $(a^2+3b^2)^3 = (a^3-9ab^2)^2 + 3(3a^2b-3b^3)^2$ directly follow.
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