Integer ratios of consecutive alternating power sums

TL;DR

This paper characterizes all pairs of positive integers where the ratio of two consecutive alternating power sums is an integer, providing a complete understanding of when such ratios are integral.

Contribution

It offers a complete characterization of all pairs (k,n) with integer ratios of consecutive alternating power sums, a novel result in the study of power sums.

Findings

Identifies all (k,n) pairs with integer ratios

Provides explicit conditions for integrality of ratios

Advances understanding of alternating power sums

Abstract

We give a characterization of all pairs $(k,n)$ of positive integers for which the ratio $$ \frac{1^k-2^k+3^k-\dots+(-1)^{n+1} n^k}{1^k-2^k+3^k-\dots+(-1)^{n}(n-1)^k} $$ of two consecutive alternating power sums is an integer.