Effective results for polynomial values of (alternating) power sums of arithmetic progressions

TL;DR

This paper establishes effective finiteness results for polynomial values of sums involving powers of arithmetic progressions and alternating sums, providing explicit bounds and conditions for such values.

Contribution

It proves new effective finiteness results for polynomial values of power sums of arithmetic progressions and their alternating variants, with explicit bounds.

Findings

Finiteness results for polynomial values of power sums

Explicit bounds for solutions involving arithmetic progressions

Effective criteria for polynomial values in these sums

Abstract

We prove effective finiteness results concerning polynomial values of the sums $$ b^k +\left(a+b\right)^k + \cdots + \left(a\left(x-1\right) + b\right)^k $$ and $$ b^k - \left(a+b\right)^k + \left(2a+b\right)^k - \ldots + (-1)^{x-1} \left(a\left(x-1\right) + b\right)^k , $$ where $a \neq 0,b, k$ are given integers with $\gcd(a,b)=1$ and $k \geq 2$.