More on the Nonexistence of Odd Perfect Numbers of a Certain Form

TL;DR

This paper proves that odd perfect numbers cannot have a certain prime factorization form when all exponents satisfy a specific modular condition, extending previous nonexistence results.

Contribution

It removes previous divisibility hypotheses and establishes nonexistence of odd perfect numbers with all exponents congruent to 2 mod 5 in their prime factorization.

Findings

No odd perfect numbers of the specified form exist under the new conditions.

The result generalizes previous nonexistence theorems.

It advances understanding of the structure of hypothetical odd perfect numbers.

Abstract

Euler showed that if an odd perfect number exists, it must be of the form $N = p^\alpha q_{1}^{2\beta_{1}}$ $\ldots$ $q_{k}^{2\beta_{k}}$, where $p, q_{1}, \ldots, q_k$ are distinct odd primes, $\alpha$, $\beta_{i} \geq 1$, for $1 \leq i \leq k$, with $p \equiv \alpha \equiv 1 \pmod{4}$. In 2005, Evans and Pearlman showed that $N$ is not perfect, if $3|N$ or $7|N$ and each $\beta_{i} \equiv 2 \pmod{5}$. We improve on this result by removing the hypothesis that $3|N$ or $7|N$ and show that $N$ is not perfect, simply, if each $\beta_{i} \equiv 2 \pmod{5}$.