The Kellner–Erdős–Moser conjecture

For positive integers $k$ and $m$, let

$$S_k(m):=1^k+2^k+\frac{\cdots}{}+(m-1)^k.$$

Kellner–Erdős–Moser conjecture. If $m\geq3$, then

$$\frac{S_k(m+1)}{S_k(m)}$$

is an integer if and only if $(m,k)=(3,1)$ or $(m,k)=(3,3)$.

This conjecture concerns divisibility properties of consecutive power sums and remains open according to the source.