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!x! where xx 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 pp such that 2p12^{p}-1 is also a prime, then (2p1)2p1(2^{p}-1)2^{p-1} will be a perfect number (Why?)

=> If there exist any natural number x>6x>6 such that x!=(2p1)2p1x!=(2^{p}-1)2^{p-1} then xx must be divisible by 3

=> xx is divisible by 3

=> (2p1)2p1(2^{p}-1)2^{p-1} is also divisible by 3

But 2p12^{p}-1 can't be divisible by 3 as it is a prime, similarly 2p12^{p-1} is only divisible by 2 not by 3

=> (2p1)2p1(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
5 months, 1 week ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

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

A Former Brilliant Member - 5 months, 1 week ago

Log in to reply

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

Zakir Husain - 5 months, 1 week ago

Log in to reply

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

A Former Brilliant Member - 5 months, 1 week ago

Log in to reply

@A Former Brilliant Member 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.

Zakir Husain - 5 months, 1 week ago

Log in to reply

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

A Former Brilliant Member - 5 months, 1 week ago

Log in to reply

@A Former Brilliant Member 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.

Zakir Husain - 5 months, 1 week ago

Log in to reply

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

A Former Brilliant Member - 5 months, 1 week ago

Log in to reply

@A Former Brilliant Member Okay, afterwards you can write the whole proof itself

Zakir Husain - 5 months, 1 week ago

Log in to reply

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

A Former Brilliant Member - 5 months, 1 week ago

Log in to reply

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

Zakir Husain - 5 months, 1 week ago

Log in to reply

What does a perfect number mean?

Mahdi Raza - 5 months, 1 week ago

Log in to reply

Nevermind, got it!

Mahdi Raza - 5 months, 1 week ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...