Forbidden integer ratios of consecutive power sums

TL;DR

This paper investigates the conjecture that the ratio of consecutive power sums is never an integer for large classes of ratios, developing techniques to exclude many potential ratios and connecting the problem to Erdős-Moser type equations.

Contribution

The paper introduces new methods to exclude many integer ratios of consecutive power sums and extends the exclusion to a large set of ratios under a conjecture on irregular primes.

Findings

Excluded ratios 3 to 1501 as ratios of consecutive power sums.

Under a conjecture, excluded a density 1 set of ratios.

Linked the problem to Erdős-Moser type equations to prove non-existence of solutions.

Abstract

Let $S_k(m):=1^k+2^k+\cdots+(m-1)^k$ denote a power sum. In 2011 Bernd Kellner formulated the conjecture that for $m\ge 4$ the ratio $S_k(m+1)/S_k(m)$ of two consecutive power sums is never an integer. We will develop some techniques that allow one to exclude many integers $\rho$ as a ratio and combine them to exclude the integers $3\le \rho\le 1501$ and, assuming a conjecture on irregular primes to be true, a set of density $1$ of ratios $\rho$. To exclude a ratio $\rho$ one has to show that the Erd\H{o}s-Moser type equation $(\rho-1)S_k(m)=m^k$ has no non-trivial solutions.