Where do odd perfect numbers live?

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...

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Bibliographic Details
Published in:arXiv.org 2018-01
Main Author: Aldi Nestor de Souza
Format: Article
Language:English
Subjects:
Number theory
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Summary: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 \( n = pm ^ 2. \)
ISSN:2331-8422