The generalised Erdős–Moser conjecture

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

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

Generalised Erdős–Moser conjecture. There are no integral solutions $(a,k,m)$ to

$$S_k(m)=am^k$$

with $m\geq2$, $k\geq2$, and $a\geq1$.

The best known results give explicit lower bounds on $m$ for hypothetical solutions, but the conjecture remains open and is described as requiring new methods.