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

TL;DR

This paper proves that for fixed integers $k,\, ext{and}\,\, ext{ell}$ with certain conditions, all solutions to a specific sum of powers equation are bounded, extending previous results to new cases.

Contribution

It establishes finiteness of solutions for the Diophantine equation with new restrictions, completing the analysis for odd $ ext{ell}$ and excluding the case $k=3$.

Findings

Solutions are bounded by an effectively computable constant $C$.

The result extends previous work by Soydan to new parameter cases.

The case for even $ ext{ell}$ was previously known and is referenced.

Abstract

Let $k,\ell\geq2$ 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 $x,y\geq1, n\geq2, k\neq3$ and $\ell\equiv 1 \pmod 2$ 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).