<s>[INST] Primitive instances where $c^3|(a^3+b^3)$ with $a,b,c\in\mathbb{N}$ It is known that by Fermat's Last Theorem there are no solutions to $a^3+b^3=c^3$ for $a,b,c\in\mathbb{N}$. I wondered about how multiplying the $c^3$ by a constant would change this fact.
Accordingly, I have been looking into instances where $c^3|(a^3+b^3)$ for $a,b\in\mathbb{N}$. In other words, solutions to the Diophantine Equation:
$a^3+b^3=dc^3$ where $a,b,$ and $c$ are pairwise co-prime and $a,b,c>0$
Obviously there are some trivial solutions. If $c=1$, for example, $a$ and $b$ can be any integers, and $d$ can be chosen to simply be $a^3+b^3$. 
By requiring that $a,b,$ and $c$ are pairwise co-prime and that $c\not=1$, we eliminate the trivial solutions, and what remains is of much more interest. 
For $a,b,c,d\le20$, there are 5 solutions:
$4^3+5^3=7*3^3$
$2^3+7^3=13*3^3$
$1^3+8^3=19*3^3$
$3^3+5^3=19*2^3$
$1^3+19^3=20*7^3$
Under $100$ there are $16$ solutions, as found by Mathematica.
My  question about this equation: Has it been studied previously? Are there infinitely many primitive solutions (which it seems like there are)? If so, can they be parametrized?
 [/INST] There are infinitely many parametric solutions for arbitrary c.
We use simple identity below.
$$(c^3n+c^3-b)^3+b^3=c^3(n+1)(c^6n^2+2c^6n-3c^3nb+3b^2+c^6-3c^3b)$$
Let $a=c^3n+c^3-b, d=(n+1)(c^6n^2+2c^6n-3c^3nb+3b^2+c^6-3c^3b)$.
Hence we get the parametric solutions of $a^3+b^3=dc^3$.
We show the examples only for $c=2,3,4,5.$ 
$$(8n+7)^3+1^3 = 2^3(n+1)(64n^2+104n+43)$$
$$(27n+26)^3+1^3 = 3^3(3n+3)(243n^2+459n+217)$$
$$(64n+61)^3+3^3 = 4^3(n+1)(4096n^2+7616n+3547)$$
$$(125n+124)^3+1^3 = 5^3(n+1)(15625n^2+30875n+15253)$$
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