A diophantine equation for sums of consecutive like powers

Simon Felten, Stefan Müller-Stach

We show that the diophantine equation $n^{\ell }+(n+1{)}^{\ell }+...+(n+k{)}^{\ell }=(n+k+1{)}^{\ell }+...+(n+2k{)}^{\ell }$ has no solutions in positive integers $k,n\ge 1$ for all $\ell \ge 3$.