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From: Archimedes Plutonium on 13 Jul 2010 15:25
While I am at it, I may as well clear out all the old unsolved Ancient
Greek conjectures
of these three:
1) Twin Primes
2) Infinitude of even Perfect Numbers
3) 1 is the only odd Perfect Number

I proved Twin Primes and even Perfect Numbers already in this thread
so may as well grapple with 1 is the only odd Perfect Number.

I did this proof in early 1990s, so it is nothing new as to the
technique
involved. I won no converts, but sometimes in mathematics a proof
acceptance
takes longer than finding a proof. People are stubborn and jeolous
like anything else.

Now the wording of this conjecture is different from the literature
for they say No
Odd Perfect number exists, but I like to use 1 as an Odd Perfect
Number and there
is no prejudice to that restatement and proof.

Now the way I prove that 1 is the only odd perfect number is that I
look upon the smallest
even perfect number of 6 and see how it is driven to be "perfect" and
I use fractions to
get me the insight.

So I see 6 as the smallest perfect even number because I see this:

1/6 + 2/6 + 3/6 = 6/6

Now that does not give me any real insight until I turn that around to
be this:

1/2 + 1/3 + 1/6 = 1

Now the insights begin to flow. I see that to ever attain "perfectness
of number"
I need 50% as one factor.

Then the major insight occurs, that the numerator is always going to
be odd
whereas the denominators are going to be a mix of odd and even.

Now do many of you readers remember the proof of the square root of 2
is
irrational and how we play around with even and odd in the proof? You
remember that
tussle back and forth of even and odd.

Well in the proof that 1 is the only odd perfect number we have a sort
of deja vu all over
again with even and odd accounting.

To be a perfect number such as 6, you need that 50% margin in one
divisor. You can
never have that 50% in a odd number. Take for example 15

1/15 + 3/15 + 5/15

1/15 + 1/5 + 1/3

So, in my proof in the early 1990s, what I was doing was saying that
if a Odd Perfect
number larger than 1 exists, it is a very strange number indeed
because it would have
to have a 50% factor and that would mean it would have to have a
denominator that was
even when denominators are odd for odd numbers.


Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
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