Odd perfect numbers do not exist. According to Euclid's formulae a perfect number is equal to $2^P(2^P-1$) where $2^P-1$ should be prime. Only $2$ is the even prime number and rest are odd. So $2^P-1$ is odd. $(2^P-1) +1= 2^P$
That is $2^P-1$ is odd , an odd number $+ 1$ is even . Therefore $2^P$ is even. Product of odd and even is always even. Since $2^P-1$ is odd and $2^P$ is even $2^P(2^P-1)$ is even. That is odd perfect numbers do not exist.
- Determining if any odd perfect numbers exist.