On equal values of products and power sums of consecutive elements in an arithmetic progression

TL;DR

This paper investigates a Diophantine equation involving sums of powers of arithmetic progression elements and products of linear factors, providing effective solutions for specific parameters and a general ineffective result using the Bilu-Tichy method.

Contribution

It offers new effective solutions for particular cases and applies the Bilu-Tichy method to derive a broad, but ineffective, result for the equation.

Findings

Effective solutions for specific parameter values

Application of Bilu-Tichy method for general results

Identification of conditions under which solutions exist

Abstract

In this paper we study the Diophantine equation \begin{align*} b^k + \left(a+b\right)^k + &\left(2a+b\right)^k + \ldots + \left(a\left(x-1\right) + b\right)^k = \\ &y\left(y+c\right) \left(y+2c\right) \ldots \left(y+ \left(\ell-1\right)c\right), \end{align*} where $a,b,c,k,\ell$ are given integers under natural conditions. We prove some effective results for special values for $c,k$ and $\ell$ and obtain a general ineffective result based on Bilu-Tichy method.