Where do odd perfect numbers live?

TL;DR

This paper investigates the structure of hypothetical odd perfect numbers, proving that if such numbers exist, their prime factorization must include a prime of the form 4k+1 with an exponent exactly equal to one.

Contribution

The paper establishes that in the prime factorization of an odd perfect number, the exponent of the prime of the form 4k+1 must be exactly one, refining the known structural constraints.

Findings

If an odd perfect number exists, it must be of the form n = p m^2 with p prime of form 4k+1 and exponent 1.

The exponent r of the prime p in the factorization is necessarily equal to 1.

This result narrows the possible forms of odd perfect numbers, contributing to the longstanding open problem.

Abstract

The existence of a perfect odd number is an old open problem of number theory. An Euler's theorem states that if an odd integer $ n $ is perfect, then $ n $ is written as $ n = p ^ rm ^ 2 $, where $ r, m $ are odd numbers, $ p $ is a prime number of the form $ 4 k + 1 $ and $ (p, m) = 1 $, where $ (x, y) $ denotes the greatest common divisor of $ x $ and $ y $. In this article we show that the exponent $ r $, of $ p $, in this equation, is necessarily equal to 1. That is, if $ n $ is an odd perfect number, then $ n $ is written as $ n = pm ^ 2. $