**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.

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:

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in`\(`

...`\)`

or`\[`

...`\]`

to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

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

Log in to reply

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

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

Log in to reply

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.

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.

Log in to reply

Certainly I will post it .just wait

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

Log in to reply