After spending sometime on this note I came to the following theorem:
There is no perfect factorial number other then 6.
Note : Throughout the proof the perfect number is assumed to be where is any natural number
For any factorial number greater than 6, it will always be an even number.
=> If any factorial number greater than 6 is a perfect number then it must be an even perfect number only
=> There must exist some prime such that is also a prime, then will be a perfect number (Why?)
=> If there exist any natural number such that then must be divisible by 3
=> is divisible by 3
=> is also divisible by 3
But can't be divisible by 3 as it is a prime, similarly is only divisible by 2 not by 3
=> can never be divisible by 3
Therefore, there is no perfect factorial number greater than 6