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If the equation of the title has an integer solution with k>=2, then m>10^{10^6}. Leo Moser showed this in 1953 by amazingly elementary methods. With the hindsight of more than 50 years his proof can be somewhat simplified. We give a further proof showing that Moser's result can be derived from a von Staudt-Clausen type theorem. Based on more recent developments concerning this equation, we derive a new result using the divisibility properties of numbers in the sequence 2^{2e+1}+1, e=0,1,2, In the final section we show that certain Erdos-Moser type equations arising in a recent paper of Kellner can be solved completely.

Moser's mathemagical work on the equation $1^k+2^k+\cdots+(m-1)^k=m^k$