
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 and positive integers 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 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 . 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 \[ \text{\seqsplit{8490421583559688410706771261086}}. \]…
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