TITLE: Can $n^4 + n^3 + n^2 + n + 1$ for $n \in \mathbb{N} \backslash \{ 0,3\}$ yield a perfect square number?
QUESTION [6 upvotes]: I tried every $n \in [4;2,000,000]$ and it seems to be true that there will be no other square numbers.
Can someone think of a proof for this conjecture?
Any kind of help will be appreciated!

REPLY [2 votes]: The solution is given in Ribenboim's book on Catalan's conjecture, where all Diophantine equations $$y^2=1+x+x^2+\cdots +x^k$$ are studied. For $k=4$, the only positive solution is $x=3$.