<s>[INST] Find all natural numbers $a,b,c$ such that $abc+ab+c=a^3$ Find all positive integers $a,b,c$ such that
$$abc+ab+c=a^3$$
My try:
Clearly $c=ak$
$abk+b+k=a^2$
$b=\frac{a^2-k}{ak+1}$ is an integer but I am not getting anything further
 [/INST] $\begin{align}
abc+ab+c &=a^3 \\
a|c, \text{ set } c=ak \hspace{2in} \\
a^2bk + a(b+k) &=a^3 \\
abk+(b+k)&=a^2 \\
a^2-abk-(b+k) &= 0
\end{align}$
$$a = \frac{bk + \sqrt{b^2k^2+4(b+k)}}{2}$$
$\to$ integer when $b^2k^2+4(b+k)$ is square. This has the same parity as $bk$ so the sum is even.

Note also that $(bk+2)^2 = b^2k^2 + 4bk+4 = b^2k^2 + 4(bk+1)$
So we need $(b+k)\geq(bk+1) \\
\Rightarrow  (b-1) \geq (bk-k) = k(b-1) \\
\Rightarrow k=1 \text{ or } b-1=0$
Therefore for any $a$, the possible answers are:


*

*$ b=1, c=(a^2-a)$

*$b= (a-1), c=a$

</s>