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

TL;DR

This paper investigates the solutions of a specific exponential Diophantine equation, proving bounds on the exponents and variables, and showing that solutions are finite and effectively computable under certain conditions.

Contribution

It establishes effective bounds on solutions to the equation involving sums of powers, extending understanding of such Diophantine equations with new finiteness results.

Findings

Solutions have bounded exponents n by an effectively computable constant.

Solutions with certain parameters are bounded in variables x, y, n by an effectively computable constant.

The results apply to equations with fixed k,l and exclude the case k=3, l even.

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 $l$.