HMMT 2020 A6

A polynomial $P(x)$ is a base-n polynomial if it is of the form $a_{d} x^{d}+a_{d-1} x^{d-1}+\cdots+a_{1} x+a_{0}$, where each $a_{i}$ is an integer between 0 and $n-1$ inclusive and $a_{d}>0$. Find the largest positive integer $n$ such that for any real number $c$, there exists at most one base- $n$ polynomial $P(x)$ for which $P(\sqrt{2}+\sqrt{3})=c$.