  The Erd\"os-Moser conjecture states that the Diophantine equation $S_k(m) =
m^k$, where $S_k(m)=1^k+2^k+...+(m-1)^k$, has no solution for positive integers
$k$ and $m$ with $k \geq 2$. We show that stronger conjectures about
consecutive values of the function $S_k$, that seem to be more naturally, imply
the Erd\"os-Moser conjecture.
