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.