Almost all entries in the character table of the symmetric group are multiples of any given prime

TL;DR

This paper proves that as the size of the symmetric group grows, almost all entries in its character table are divisible by any fixed prime, confirming a conjecture by Miller.

Contribution

It establishes that nearly all character table entries of symmetric groups are divisible by any fixed prime as the group size increases, resolving Miller's conjecture.

Findings

Almost all entries are divisible by the fixed prime for large N.

The result holds for any fixed prime as N approaches infinity.

It confirms Miller's conjecture about divisibility in character tables.

Abstract

We show that almost every entry in the character table of $S_N$ is divisible by any fixed prime as $N\to\infty$. This proves a conjecture of Miller.