Port Fillings for Primary Pseudoperfect Numbers

TL;DR

This paper investigates primary pseudoperfect numbers through a novel local residual equation framework, providing new insights into their structure and the question of their infinitude.

Contribution

It introduces a local language for residual equations and a port-based filling method to analyze primary pseudoperfect numbers, advancing understanding beyond known finite cases.

Findings

Established a port filling framework for primary pseudoperfect numbers.

Separated inherited and primitive fillings using the product rule for arithmetic derivatives.

Provided unconditional results on the structure and properties of these numbers.

Abstract

Erd\H{o}s asked whether there are infinitely many finite sets of distinct primes $p_1<\cdots<p_k$ and positive integers $m$ such that \begin{equation}\label{eq:erdos-original} \frac1{p_1}+\cdots+\frac1{p_k}=1-\frac1m. \end{equation} This is Erd\H{o}s Problems \#313~\cite{ErdosProblems313}. As recalled below, it is equivalent to the infinitude of primary pseudoperfect numbers. Following Butske, Jaje, and Mayernik~\cite{ButskeJajeMayernik}, a squarefree positive integer $n$ is a \emph{primary pseudoperfect number} if \begin{equation}\label{eq:ppn-def} \frac1n+\sum_{p\mid n}\frac1p=1, \end{equation} where the sum is over the prime divisors of $n$. OEIS A054377~\cite{OEISA054377} records the initial values \[ \begin{array}{c} 2,\ 6,\ 42,\ 1806,\ 47058,\\[2pt] 2214502422,\ 52495396602. \end{array} \] and the eight-prime-factor example \[ \text{\seqsplit{8490421583559688410706771261086}}. \] Butske, Jaje, and Mayernik proved by computation that for each $r\le 8$ there is exactly one primary pseudoperfect number with $r$ distinct prime factors~\cite{ButskeJajeMayernik}. This result gives a useful baseline, but it does not address later layers or the infinitude problem. This paper uses a local language for residual equations. A \emph{port} is a pair $(R,c)$, and a squarefree integer $B$ fills it if \[ \Delta_{R,c}(B):=cB-R\partial(B)=1. \] The corresponding reciprocal form is \[ \sum_{q\mid B}\frac1q+\frac1{RB}=\frac cR. \] The product rule for the arithmetic derivative gives the composition law for ports. This law separates fillings inherited from smaller primary pseudoperfect numbers from fillings that are primitive relative to the fixed residual equation. The unconditional results of the paper are as follows.