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.
http://mathworld.wolfram.com/UnsolvedProblems.html
- Determining if any odd perfect numbers exist.
You're wrong. All that Euclid proved was that when $2^p-1$ is prime, then $2^{p-1}(2^p-1)$ is perfect. He never said that all perfect numbers can be obtained by this process.
In the XVIIIth century, Euler proved that every even perfect number can be obtained by that process. He never stated that this holds for odd perfect numbers too.