Filaseta–Moy conjecture on the Galois groups of the sextic binomial polynomials

Let

$$P_{n,6}(X)=\binom{n}{0}+\binom{n}{1}X+\cdots+\binom{n}{6}X^6$$

be the degree-six truncation of the binomial expansion, and let $\mathfrak{S}_6$ denote the full symmetric group on six symbols. Filaseta–Moy conjecture. The Galois group of $P_{n,6}$ is isomorphic to $\mathfrak{S}_6$ for all $n\geq 11$. This conjecture refines the result that only $O(\log N)$ values of $n\leq N$ can have a smaller Galois group; numerical computations up to $10^{10}$ suggested that the known exceptional value $n=10$ is the last one.