Are there any $n$ for which $ n^4+n^3+n^2+n+1$ is a perfect square? Are there any positive $n$ for which $ n^4+n^3+n^2+n+1$ is a perfect square?
I tried to simplify 
\begin{align*}
n^4+n^3+n^2+n+1 &= n^2(n^2+1)+n(n^2+1)+1\\
 &= (n^2+n)(n^2+1)+1 \\
  &= n(n+1)(n^2+1)+1
\end{align*}
Then I assumed that the above expression is a square; then
$$ n(n+1)(n^2+1)+1 = k^2$$
$$ 
\begin{align*}
n(n+1)(n^2+1) &= (k^2-1) \\
   &= (k+1)(k-1)
\end{align*}
$$ 
Then trying to reason with prime factors, but cannot find a concrete proof yet.


Here is my answer!
H. Bensom, Germany
