On perfect factorial numbers

After spending sometime on this note I came to the following theorem:

There is no perfect factorial number other then 6.

Proof:

Note : Throughout the proof the perfect number is assumed to be $x!$ where $x$ 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 $p$ such that $2^{p}-1$ is also a prime, then $(2^{p}-1)2^{p-1}$ will be a perfect number (Why?)

=> If there exist any natural number $x>6$ such that $x!=(2^{p}-1)2^{p-1}$ then $x$ must be divisible by 3

=> $x$ is divisible by 3

=> $(2^{p}-1)2^{p-1}$ is also divisible by 3

But $2^{p}-1$ can't be divisible by 3 as it is a prime, similarly $2^{p-1}$ is only divisible by 2 not by 3

=> $(2^{p}-1)2^{p-1}$ can never be divisible by 3

Therefore, there is no perfect factorial number greater than 6

#NumberTheory

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