# Primary Pseudoperfect Numbers, Arithmetic Progressions, and the Erd\H{o}s-Moser Equation

## TL;DR

This paper explores properties of primary pseudoperfect numbers, discovers an arithmetic progression in their residues, and proposes a conjecture that implies a new lower bound on solutions to the Erdős-Moser equation.

## Contribution

It establishes new modular properties of PPNs, identifies an arithmetic progression in their sequence, and links these findings to a conjecture that significantly raises the lower bound for solutions of the Erdős-Moser problem.

## Key findings

- PPNs satisfy specific congruences modulo 36 when divisible by 6
- A 7-term arithmetic progression of PPN residues modulo 384 is identified
- A conjecture is proposed that implies a lower bound >10^{3.99×10^{20}} for Erdős-Moser solutions

## Abstract

A primary pseudoperfect number (PPN) is an integer $K > 1$ such that the reciprocals of $K$ and its prime factors sum to 1. PPNs arise in studying perfectly weighted graphs and singularities of algebraic surfaces, and are related to Sylvester's sequence, Giuga numbers, Zn\'am's problem, the inheritance problem, and Curtiss's bound on solutions of a unit fraction equation. Here we show $K \equiv 6 \pmod{6^2}$ if $6\mid K$, and uncover a remarkable $7$-term arithmetic progression of residues modulo $6^2\cdot8$ in the sequence of known PPNs. On that basis, we pose a conjecture which leads to a conditional proof of the new record lower bound $k>10^{3.99\times10^{20}}$ on any non-trivial solution to the Erd\H{o}s-Moser Diophantine equation $1^n + 2^n + \dotsb + k^n = (k+1)^n$.