Question 128
The teacher wishes to compose a problem for his students of the following form. He tells them that he thinks of a monic polynomial of degree $2017$ with integer coefficients. Then he tells them $k$ integers $n_1, \dots, n_k$, and finally he tells them the value of the product $P(n_1) \cdot P(n_2) \dots P(n_k)$. By all this data, the students need to find a polynomial satisfying the described conditions. Determine the least value of $k$ for which the teacher can compose a problem of this form so that any suitable polynomial found by the students will necessarily coincide with the polynomial he thinks of.