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 $n=p^r{m}^2$, where $r,m$ are odd numbers, $p$ is a prime number of the form $4k+1$ 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. $