An open conjecture of Erdős and Moser (from around 1950) is that the only solution of the Diophantine equation

${1}^{n}+{2}^{n}+\cdots +{k}^{n}={(k+1)}^{n}$ is the trivial solution

$1+2=3$.

*Y. Gallot, P. Moree* and

*W. Zudilin* [Math. Comput. 80, No. 274, 1221–1237 (2011;

Zbl 1231.11038)] showed that if there is a further solution then both

$k$ and

$n$ must exceed

${10}^{{10}^{9}}$. By reducing the equation modulo

${k}^{2}$ the authors find some new conditions that solutions

$(k,n)$ have to satisfy. The proofs use divisibility properties of power sums as well as Lerch’s relation between Fermat and Wilson quotients.