# 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

Note by Zakir Husain
9 months, 2 weeks ago

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@Zakir Husain, should we collaborate together and publish this on Wikipedia? Just a thought...

- 9 months, 2 weeks ago

I haven't signed in Wikipedia, if you are then you can

- 9 months, 2 weeks ago

I just created an account - should I mention you and do you mind if I 'borrow' your proof for this page - we will need it for proof that this is unique. @Zakir Husain

- 9 months, 2 weeks ago

What you mean by 'borrow', you can mention the proof in the references using the link of this notice. You should mention this page and me also.

- 9 months, 2 weeks ago

I did, but I still need the LaTeX of this proof because I still wish to show this on the article.

- 9 months, 2 weeks ago

You can write the proof in the article itself but don't forget to mention this note and me, also mention them in the article.

- 9 months, 2 weeks ago

I mentioned the note and you in the article (currently a draft)

- 9 months, 2 weeks ago

Okay, afterwards you can write the whole proof itself

- 9 months, 2 weeks ago

I have submitted it for review - hopefully, they'll clear up the LaTeX issues, but we've got to wait up to 5 weeks.

- 9 months, 2 weeks ago

@Yajat Shamji If you can then Try to prove it!

- 9 months, 2 weeks ago

What does a perfect number mean?

- 9 months, 2 weeks ago

Nevermind, got it!

- 9 months, 2 weeks ago