Perfect numbers

[Olivier Gérard]

Simon was asking for more reference.
There is a detailed history of the search for perfect numbers at

http://www-groups.dcs.st-andrews.ac.uk/~history/HistTopics/Perfect_numbers.html

You will learn another Euler result. An odd perfect number if it exists
has the form

(4n+1)^(4k+1) b^2   with 4n+1 prime.


Olivier


On Wed, Apr 18, 2001 at 10:24:57AM -0400, Joe Crump wrote:
> Doesn't this just stem from Euler's formula (2^k - 1) * 2^(k-1)?
> 
> Obviously the binary rep is going to be a sequence of ones
> (the 2^k - 1 term) shifted left k-1 times (the 2^(k-1) term).
> 
> The number of ones is prime because k needs to be prime.
> 
> - Joe
> 
> -----Original Message-----
> From: Simon Colton [mailto:simonco at dai.ed.ac.uk] 
> Sent: Wednesday, April 18, 2001 10:14 AM
> To: seqfan at ext.jussieu.fr
> Subject: 
> 
> 
> Dear SeqFans,
> 
> I thought you might be interested in another cute theorem produced by my
> HR program recently.
> 
> I imagine that this, or a more general, result is well known, and would
> very much appreciate any references.
> 
> Cheers,
> 
> Simon
> ---
> http://www.dai.ed.ac.uk/~simonco
>