Nonexistence of Odd Perfect Numbers

14 December 2023, Version 1
This content is an early or alternative research output and has not been peer-reviewed by Cambridge University Press at the time of posting.

Abstract

Let $\sigma(n)$ denote the sum-of-divisors function $\sigma(n)=\sum_{d \mid n} d$. We also define $I(n) = \frac{\sigma(n)}{n}$ as the abundancy index of $n$. An integer $n$ is perfect if $I(n)=2$. It is unknown whether any odd perfect number exists or not. In this note, using the properties of the abundancy index function, we prove the non-existence of odd perfect numbers.

Keywords

Elementary Number Theory
prime numbers
odd perfect numbers
sum-of-divisors function
abundancy index function

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