Waste less time on Facebook — follow Brilliant.
×

Superperfect Numbers form

\(n\) is called a superperfect number when \(\sigma(\sigma(n))=2n\).I saw every superperfect numbers is in the form \(2^k\) when \(2^{k+1}-1\) is a Mersenne prime.I can prove that numbers in this form will be superperfect but cannot prove the converse that all superperfect numbers are in this form. can anybody help me to prove? If this is not true then also tell me I am not sure about it.

Note by Kalpok Guha
2 years, 9 months ago

No vote yet
1 vote

  Easy Math Editor

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 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

Sort by:

Top Newest

Can you please provide a proof that if \(2^{k+1} -1 \) is a Mersenne prime then \(2^{k}\) is a perfect number because that might help us when we try to prove the converse.

Curtis Clement - 2 years, 9 months ago

Log in to reply

Here let \(n=2^k\)

\(\sigma(\sigma(2^k)\)

=\(\sigma(\frac {2^{k+1}-1}{2-1})\)

=\(\sigma(2^{k+1}-1)\)

As \(2^{k+1}-1\) is a prime \(\sigma(2^{k+1}-1)=(2^{k+1}-1)+1=2^{k+1}=2.2^k=2n\)

Proved that it is always a superperfect number.

Kalpok Guha - 2 years, 9 months ago

Log in to reply

Certainly I will post it .just wait

Kalpok Guha - 2 years, 9 months ago

Log in to reply

This might be helpful...check out perfect Numbers, as explained by Numberphile: (http://m.youtube.com/watch? v=q8n15q1v4Xo)

B.S.Bharath Sai Guhan - 2 years, 9 months ago

Log in to reply

Thank you very much it became really useful to me .Though here they only proved the form of perfect number but listening to the class I got idea to the form of superperfect number.Identified the series of super perfect number

Kalpok Guha - 2 years, 9 months ago

Log in to reply

I read it on Elementary Number Theory by David Burton,then opened wiki and saw but there is no proof there neither direct one nor converse

Kalpok Guha - 2 years, 9 months ago

Log in to reply

You mean you saw it on wiki? 'Cause its there...

Danny Kills - 2 years, 9 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...