<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] Using the hint that OP stated,
$$ \begin{array} { l l}
(a^2 + 3b^2 )^3 & = a^6 + 9 a^4 b^2 + 27a^2 b^4 + 27b^6 \\
& = (a^6 + 6 a^4 b^2 + 9 a^2 b^4) + 3( a^4 b^2 + 6a^2 b^4 + 9b^6 ) \\
& = (a^3 + 3ab^2)^2 + 3 (a^2b + 3b^3)^2 \end{array}$$
</s>