ELMO 2023 Problem 5

Find the least positive integer $M$ for which there exist a positive integer $n$ and polynomials $P_1(x)$, $P_2(x)$, $\ldots$, $P_n(x)$ with integer coefficients satisfying $$Mx=P_1(x)^3+P_2(x)^3+\cdots+P_n(x)^3.$$

Proposed by Karthik Vedula