The Diophantine Equation $(x+1{)}^k+(x+2{)}^k+\cdots +(\ell x{)}^k=y^n$ Revisited

Abstract

Let $k,\ell \ge 2$ be fixed integers and $C$ be an effectively computable constant depending only on $k$ and $\ell$. In this paper, we prove that all solutions of the equation $(x+1{)}^k+(x+2{)}^k+...+(\ell x{)}^k=y^n$ in integers $x,y,n$ with $x,y\ge 1,n\ge 2,k\ne 3$ and $\ell \equiv 1(\mathrm{mod}2)$ satisfy $\max \{x,y,n\}<C$. The case when $\ell$ is even has already been completed by Soydan (Publ. Math. Debrecen 91 (2017), pp. 369-382).

arXiv comment8 pages, accepted for publication

References in corpus (1)

• On perfect powers that are sums of cubes of a three term arithmetic progression