Question 143
Find all pairs of integers $(c, d)$, both greater than $1$, such that the following holds:
For any monic polynomial $Q$ of degree $d$ with integer coefficients and for any prime $p \gt c(2c+1)$, there exists a set $S$ of at most $\left(\frac{2c-1}{2c+1}\right)p$ integers, such that
$$
\bigcup_{s \in S} \{s, Q(s), Q(Q(s)), Q(Q(Q(s))), \ldots\}
$$
is a complete residue system modulo $p$ (i.e., intersects with every residue class modulo $p$).
For any monic polynomial $Q$ of degree $d$ with integer coefficients and for any prime $p \gt c(2c+1)$, there exists a set $S$ of at most $\left(\frac{2c-1}{2c+1}\right)p$ integers, such that
$$
\bigcup_{s \in S} \{s, Q(s), Q(Q(s)), Q(Q(Q(s))), \ldots\}
$$
is a complete residue system modulo $p$ (i.e., intersects with every residue class modulo $p$).