In 1956 J. J. Schäffer proved that the equation $$1^k+2^k+\cdots+x^k=y^n$$ for fixed integers $k\ge1,n\ge2$ has only finitely many solutions in positive integers unless $(k,n)\in\{(1,2),(3,2),(3,4),(5,2)\}$ [see Theorem 2, p. 27 in the file's numbering of pages]. Furthermore, he conjectured the equation has only trivial solution $x=y=1$ except $(k,n,x,y)=(2,2,24,70)$ (and excluding those four pairs of $(k,n)$).

I was able to find partial results, that the conjecture is true for

  1. $k\le11$ [see this]
  2. $n=2,k\le58$ [mentioned there as well]
  3. $k<170$ odd and even $n$ [see this one]

The newest paper (3.) was published in 2007 and I didn't find newer articles. So here is my question, is Schäffer's conjecture still an open problem?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.