  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. $
