On stronger conjectures that imply the Erd\"os-Moser conjecture

TL;DR

This paper explores stronger conjectures related to the sum of powers function that, if true, would imply the Erd"os-Moser conjecture, providing a new approach to this longstanding problem.

Contribution

It introduces stronger conjectures about consecutive values of the sum of powers function that imply the Erd"os-Moser conjecture, offering a novel perspective on the problem.

Findings

Stronger conjectures about $S_k$ imply the Erd"os-Moser conjecture.

Connecting properties of consecutive sums to the conjecture.

Provides a new pathway for approaching the Erd"os-Moser conjecture.

Abstract

The Erd\"os-Moser conjecture states that the Diophantine equation $S_k(m) = m^k$, where $S_k(m)=1^k+2^k+...+(m-1)^k$, has no solution for positive integers $k$ and $m$ with $k \geq 2$. We show that stronger conjectures about consecutive values of the function $S_k$, that seem to be more naturally, imply the Erd\"os-Moser conjecture.