On the Diophantine equation $(x+1)^{k}+(x+2)^{k}+...+(lx)^{k}=y^{n}$

Abstract: Let $k,l\geq2$ be fixed integers. In this paper, firstly, we prove that all solutions of the equation $(x+1)^{{k}+(x+2){k}+...+(lx){k}=y{n}$} in integers $x,y,n$ with $x,y\geq1, n\geq2$ satisfy $n<C_{1}$ where $C_{1}=C_{1}(l,k)$ is an effectively computable constant. Secondly, we prove that all solutions of this equation in integers $x,y,n$ with $x,y\geq1, n\geq2, k\neq3$ and $l\equiv0 \pmod 2$ satisfy $\max{x,y,n}<C_{2}$ where $C_{2}$ is an effectively computable constant depending only on $k$ and $l$.