Find all integers $x$ such that $x^{4}+x^{3}+x^{2}+x+1$ is a perfect square.
We are given a potential solution $x$. Let's then bound $x^{4}+x^{3}+x^{2}+x+1$ between two consecutive squares as in Exercise 2. We notice that $A=4\left(x^{4}+x^{3}+x^{2}+x+1\right)$ is still a perfect square and:

$$
\left(2 x^{2}+x\right)^{2}=4 x^{4}+4 x^{3}+x^{2}0$ (always true) and $x^{2}-2 x-3=(x-3)(x+1)>0$.

Necessarily, we have $x \in\{-1 ; 0 ; 1 ; 2 ; 3\}$.

By testing the 5 cases by hand, we obtain the only solutions $-1,0,3$.