>............................2*t...........t - 2

>........................._________......_________

>.........................\..............\

>..........................\..............\

> N=(2^t)*(2^(t +1) -1) =...}..2^r...-.....}..2^s....; (t +1)= p

> from a Mersenne prime..../............../

>........................./________....../________

>............................r= 1...........s= 1

> ...

--- In primenumbers@yahoogroups.com, "leavemsg1" <leavemsg1@...> wrote:

>

> Hi, Dr. Smith,

>

> I got it!!!!!!!!!!!!!!!... here it is...

> you can visit my website... www.oddperfectnumbers.com.

> or read below.

> ...

> for those of you that have visited my website, I've spent

> the last 10 years (almost 100 drafts) searching for a rea-

> sonable explanation for why odd-perfect numbers don't exist,

> and here it is...

> ...

> a puzzle of antiquity, fittingly solved, if you will...

> using simple algebra.

> ...

> the first few 'perfect' numbers are 6, 28, 496, and 8128.

> Euler proved that they were all of the form... N= (2^(n-1))

> *(2^n -1), he knew that the (2^n -1) portion had to always

> be prime, and he also noted that sigma(N) = 2N when N is a

> 'perfect' number. Notice how I'm putting the quotations

> around the word 'perfect'.

> ...

> the situation was more perfect than you might imagine, and

> I'm sure that you will be convinced that Euler's famous form-

> ula cannot be altered as he mentioned in order to produce an

> odd-perfect number! Here's what was missing...

>............................2*t...........t - 2

>........................._________......_________

>.........................\..............\

>..........................\..............\

> N=(2^t)*(2^(t +1) -1) =...}..2^r...-.....}..2^s....; (t +1)= p

> from a Mersenne prime..../............../

>........................./________....../________

>............................r= 1...........s= 1

> ...

> (OR) 6 = 2^1 +2^2 -(0),

> 28 = 2^1 +2^2 +2^3 +2^4 -(2^1),

> 496 = 2^1 +2^2 +2^3 +2^4 +2^5 +2^6 +2^7 +2^8 - (2^3 +2^2 +2^1),

> 8128 = etc.

> ...

> when 'N' is 'perfect', 'N' is equal to the sum (or rather the

> difference) of two summations that are purely binary. It was

> more delicate that previously thought. Changing the base from

> '2' to something else would be the only meaningful change to

> such a demanding, connective equation, and that leads us to the

> conclusion that an 'odd-perfect-number notion' is out of the

> question, and can't be 'perfect'.

> ...

> thus, odd-perfect numbers don't exist!

> ...

> obviously, when someone as prolific as Euler stated that this

> problem was too complicated to be easily solved, every mathema-

> tician was easily pursuaded to believe it; but if you (OR) I

> decided to retain Euler's already-proven formula for an odd-

> perfect number, it's surprising to me that we could have been

> mislead by his prior conclusion!

> ...

> *QED

> Bill Bouris

> 12/5/2011

> ...

>