What are integers $x$ and $n$ such as $ x^4+x^3+x^2+x+1 = n^2$?

Question

What are the integer numbers $x$ such as $x^4+x^3+x^2+x+1$ is the square of an integer?

$0$ and $-1$ are obvious answers, but how many are there?

Answer 1

Idea: Bound between known perfect squares.

For $ x \neq 0$, we have (expand and verify) $$ ( x^2 + \frac{x}{2} ) ^ 2 < x^4 + x^3 + x^2 + x + 1 < (x^2 + \frac{x}{2} + 1)^2. $$

Hence, the only possibility for a perfect square is for $x$ to be odd, and the square is equal to $ ( x^2 + \frac{x}{2} + \frac{1}{2})^2$. This gives us (expand and simplify) $ \frac{1}{4} (-x^2 + 2x + 3) =0$, hence $x=-1, 3$.

Adding in $x=0$, we get 3 solutions.