TITLE: Finding $n$ such that $\frac{n^4 + 1}{n^2 +n + 1}$ is an integer
QUESTION [6 upvotes]: Find all integers such that $\dfrac{n^4 + 1}{n^2 +n + 1}$ is an integer. 

I have no idea how to solve things like this and what i tried to do didn't get me anywhere. I'd be grateful for any help!

REPLY [2 votes]: Doing the elementary polynomial division we get:
$\frac{n^4 + 1}{n^2 +n + 1}=n^2-n+ \frac{n+1}{n^2+n+1}$
We do not need to consider now $n^2-n$ since it is always an integer(for integers $n$). Then observe $|\frac{n+1}{n^2+n+1}|\leq1$. So we do not have many chances that it will be an integer; it is $-1$, $0$, $1$. Write each equality case explicitly and see easily that the solutions are: $n=0$ and $n=-1$.