On even entries in the character table of the symmetric group

We show that almost every entry in the character table of $S_n$ is even as $n\to\infty$. This resolves a conjecture of Miller. We similarly prove that almost every entry in the character table of $S_n$ is zero modulo $3,5,7,11,$ and $13$ as $n\to\infty$, partially addressing another conjecture of Miller.