On the unsolvability of certain equations of Erd\H{o}s-Moser type

TL;DR

This paper investigates the unsolvability of certain equations related to power sums, providing evidence that specific ratios of these sums are never integers, and proposes related conjectures about their properties.

Contribution

The paper extends the study of Erdős-Moser type equations by proving non-existence of solutions for equations involving odd power sums for many integers, and introduces new conjectures on ratios of these sums.

Findings

No solutions for the equation aT_k(m)=(2m+1)^k for many integers a.

Supports the conjecture that T_k(m+1)/T_k(m) is never an integer for m>1.

Provides partial results towards the unsolvability of related power sum equations.

Abstract

Let $S_k(m):=\sum_{j=1}^{m-1}j^k$ denote a power sum. In 2011, Kellner proposed the conjecture that for $m>3$ the ratio $S_k(m+1)/S_k(m)$ is never an integer, or, equivalently, that for any positive integer $a$, the equation $aS_k(m)=m^k$ has no solutions in positive integers $k$ and $m$ with $m>3$. In this paper, we show that for many integers $a$ the equation $aT_k(m)=(2m+1)^k$, where $T_k(m):=\sum_{j=1}^m(2j-1)^k$, has no solutions in positive integers $k$ and $m$. This leads us to the conjecture that for $m>1$ the ratio $T_k(m+1)/T_k(m)$ is never an integer.