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An open conjecture of Erdos and Moser is that the only solution of theDiophantine equation in the title is the trivial solution 1+2=3. Reducing theequation modulo k and k^2, we give necessary and sufficient conditions onsolutions to the resulting congruence and supercongruence. A corollary is a newproof of Moser's result that the conjecture is true for odd exponents n. Wealso connect solutions k of the congruence to primary pseudoperfect numbers andto a result of Zagier. The proofs use divisibility properties of power sums aswell as Lerch's relation between Fermat and Wilson quotients.

Reducing the Erdos-Moser equation 1^n + 2^n + . . . + k^n = (k+1)^n modulo k and k^2