In my other question two days ago I asked for confirmation, whether one step of an attempt to that problem
does there exist $k,n>2$ , such that $\sum_{j=1}^k j^n = (k+1)^n $ ?
was justified. I had fiddled with that problem some years ago (2006/2007) and attributed this problem to Paul Erdős since - I think because of some informal comment of some discutant whom I took as authority (so I missed the point to track this down to a source at that time of exploration...).
After some search in my own text-sources and now googling for online sources I don't find any useful hint for the authorship of this problem.
Could someone kindly provide a reference or a hint for that problem/context and its author?
Guy, Unsolved Problems In Number Theory, 3rd edition, Problem D7: Sum of consecutive powers made a power. "Rufus Bowen conjectured that the equation $$1^n+2^n+\cdots+m^n=(m+1)^n$$ has no nontrivial solutions...." Guy gives a lot more info on what's known about the question, but doesn't give a reference for Bowen. The earliest reference he gives with the equation in the title is L Moser, On the diophantine equation $1^n+2^n+\cdots+(m-1)^n=m^n$, Scripta Math 19 (1953) 84-88, MR 14, 950. The only older reference he gives is P Erdős, Advanced problem 4347, Amer Math Monthly 56 (1949) 343. I haven't looked at either one.