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Reducing the Erdős-Moser equation $$1^n + 2^n + \dots + k^n = (k + 1)^n$$ 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 $$1^n+2^n+\dots+k^n=(k+1)^n$$ 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.

##### MSC:
 11D61 Exponential Diophantine equations 11D79 Congruences in many variables 11A41 Primes
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