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 $(p, m) = 1$, 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 $n = pm ^ 2.$