# Source of the problem: does there exist k,n>2, such that $\sum_{j=1}^k j^n = (k+1)^n$?

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?

• I know there was a version of this problem in art and craft of problem solving. – Mark May 2 '11 at 16:02
• @Mark: Thanks for the hint. I found a page related to the book online at wiley; but don't see the possibility to browse meaningfully to find it. Do you have a more specific reference? – Gottfried Helms May 2 '11 at 17:08
• Unfortunately I don't have the book with me anymore but I do remember a question that asked about this problem but I suspect that the author didn't have a reference of the original source either. – Mark May 2 '11 at 21:00

## 1 Answer

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.

• Thanks, that clears the matter. I'll look up your references. – Gottfried Helms May 3 '11 at 1:00
• That linked also to some articles of J.van de Lune/te Riele in the 1970ies available from the CWI, Amsterdam. – Gottfried Helms May 3 '11 at 2:28
• The last number of the Monthly (april 2011) carries an article by Pieter Moree on this subject. – Julián Aguirre May 3 '11 at 13:57
• @Julián: Thank you very much, just could take a look at it. From this I found also, that Pieter Moree has several discussions of this topic, available at the Max Planck Institute of Mathematics in Bonn - interesting. – Gottfried Helms May 3 '11 at 16:04