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

Abstract

Let $k,l\ge 2$ 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 $x,y\ge 1,n\ge 2$ 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 $x,y\ge 1,n\ge 2,k\ne 3$ and $l\equiv 0(\mathrm{mod}2)$ satisfy $\max \{x,y,n\}<C_2$ where $C_2$ is an effectively computable constant depending only on $k$ and $l$.

13 pages, to appear

Cited by in corpus (2)

• On the Diophantine equation $(x+1{)}^k+(x+2{)}^k+...+(2x{)}^k=y^n$
• On perfect powers that are sums of cubes of a five term arithmetic progression