Question 94
Emile has created an exercise for Benoît. He tells him that he has secretly chosen a monic polynomial $P$ of degree $2023$ with integer coefficients, that is, of the form
$$
P(X) = X^{2023} + a_{2022} X^{2022} + a_{2021} X^{2021} + \ldots + a_{1} X + a_{0}
$$
where $a_{0}, a_{1}, \ldots, a_{2022}$ are integers. He gives Benoît $k$ integers $n_{1}, n_{2}, \ldots, n_{k}$, where $k$ is a positive integer, together with the value of the product $P\left(n_{1}\right) P\left(n_{2}\right) \ldots P\left(n_{k}\right)$. From this knowledge, Benoît must try to recover the polynomial $P$.
Find the minimal integer $k$ such that Emile can find $P$ and $n_{1}, \ldots, n_{k}$ ensuring that the only polynomial consistent with the information given to Benoît is $P$.
$$
P(X) = X^{2023} + a_{2022} X^{2022} + a_{2021} X^{2021} + \ldots + a_{1} X + a_{0}
$$
where $a_{0}, a_{1}, \ldots, a_{2022}$ are integers. He gives Benoît $k$ integers $n_{1}, n_{2}, \ldots, n_{k}$, where $k$ is a positive integer, together with the value of the product $P\left(n_{1}\right) P\left(n_{2}\right) \ldots P\left(n_{k}\right)$. From this knowledge, Benoît must try to recover the polynomial $P$.
Find the minimal integer $k$ such that Emile can find $P$ and $n_{1}, \ldots, n_{k}$ ensuring that the only polynomial consistent with the information given to Benoît is $P$.