Let $P$ be a polynomial with integer coefficients such that: $P(-1)=-4, P(-3)=-40$ and $P(-5)=$
$-156$.

What is the maximum number of integers $x$ for which: $P(P(x))=x^{2}$?
On a : $P(x) \equiv P(x \bmod 3) \bmod 3$ for all integer $x$. For each of the cases $x \equiv 0,1,2 \bmod 3$, we find $P(P(x)) \equiv 2 \bmod 3$ but a square is congruent to 0 or $1 \bmod 3:$ there are therefore none.