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Reducing the Erdős-Moser equation ${1}^{n}+{2}^{n}+\cdots +{k}^{n}={\left(k+1\right)}^{n}$ modulo $k$ and ${k}^{2}$. (English) Zbl 1233.11038
An open conjecture of Erdős and Moser (from around 1950) is that the only solution of the Diophantine equation ${1}^{n}+{2}^{n}+\cdots +{k}^{n}={\left(k+1\right)}^{n}$ 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 $\left(k,n\right)$ have to satisfy. The proofs use divisibility properties of power sums as well as Lerch’s relation between Fermat and Wilson quotients.
##### MSC:
 11D61 Exponential diophantine equations 11D79 Congruences in many variables 11A41 Elementary prime number theory