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From: Archimedes Plutonium on 14 Jul 2010 02:08


Archimedes Plutonium wrote:
> Archimedes Plutonium wrote:
> > While I am at it, may as well jogg the memory of how sqrt2 is proven
> > irrational as a tug of
> > war between being even and odd:
> > --- quoting from Wikipedia ---
> > Assume that √2 is a rational number, meaning that there exists an
> > integer a and an integer b in general such that a / b = √2.
> >
> > Then √2 can be written as an irreducible fraction a / b such that a
> > and b are coprime integers and (a / b)2 = 2.
> >
> > It follows that a2 / b2 = 2 and a2 = 2 b2.   ( (a / b)n = an / bn  )
> >
> > Therefore a2 is even because it is equal to 2 b2. (2 b2 is necessarily
> > even because it is 2 times another whole number; that is what "even"
> > means.)
> >
> > It follows that a must be even (as squares of odd integers are
> > themselves odd).
> >
> > Because a is even, there exists an integer k that fulfills: a = 2k.
> >
> > Substituting 2k from (6) for a in the second equation of (3): 2b2 =
> > (2k)2 is equivalent to 2b2 = 4k2 is equivalent to b2 = 2k2.
> >
> > Because 2k2 is divisible by two and therefore even, and because 2k2 =
> > b2, it follows that b2 is also even which means that b is even.
> >
> > By (5) and (8) a and b are both even, which contradicts that a / b is
> > irreducible as stated in (2).
> >
> > --- end quoting Wikipedia on sqrt2 irrational proof ---
> >
> > Now remember that most people define Perfect Number such as 6 with the
> > factor of 2 as
> > in this:
> >
> > 1/6 + 2/6 + 3/6 + 6/6 = 2
> >
> > whereas I define it as a factor of 1:
> >
> > 1/6 + 2/6 + 3/6 = 1
> >
> > I do it that way so as to allow me to say that 1 is the only odd
> > perfect number.
> >
> > And the reason I bring this up is to show you that there are an even
> > number
> > of factors of 4 of them compared when = 2, to an odd number of factors
> > when = 1.
> >
> > So when we add 6/6 we have an even number of factors in the equation
> > whereas when
> > we delete 6/6 we have an odd number of factors. This is important in
> > the proof, because
> > to have existence of even numbered perfect numbers depends on one of
> > them being
> > 50% and thus making the rest of the factors an even number to join up
> > to fill in for the
> > other 50% needed to be perfect. Whereas in odd perfect numbers, we
> > have an odd number
> > of factors in the summation for there is never a 50% factor that we
> > can eliminate out.
> >
> > The only odd perfect number that could ever be mustered would be one
> > in which looks like this:
> >
> > 33.33...% + 33.333....% + 33.333....% but that case is impossible
> > since you cannot have
> > three summations all of the same percentage.
> >
> > Now that maybe a proof in itself that no odd perfect number other than
> > 1 exists. To argue that
> > to have a odd perfect number the outcome must devolve into 1/3 + 1/3 +
> > 1/3 for the outcome surely cannot devolve into 50% + (summing of
> > another 50%)
> >
> > Archimedes Plutonium wrote:
> > > 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.
> > >
> >
> > So what I argued in my earlier 1990s proof that 1 is the only odd
> > perfect number is that
> > much the same as square root of 2 as rational is impossible since it
> > then destroys the meaning of odd versus even factorability.
> >
> > In order to have a Odd Perfect Number larger than 1, would entail
> > either one of these
> > two impossible situations:
> >
> > (a) we have 1/3 + 1/3 + 1/3
> > or
>
> I was typing too fast, let me correct that. I should have included:
>
> 1/3 + 1/3combo + 1/3combo
>
> In the case of even perfect numbers we have always a 50% and then it
> is just a matter of adding up the other 50% to get perfect so I wrote
> 1/2 + 1/2combo for even perfect numbers
>
> But to get a perfect odd number we can never have the 50% so we must
> have something like a
>
> 1/3 + 1/3combo + 1/3combo
> I think I can shorten the proof by noting why that is impossible.
>
> Now I do remember a expert in this field reporting about what he
> called, correct me if wrong,
> about the surplus and deficit of numbers in contention for
> perfectness. What he meant was that there are even numbers that are
> below being perfect and then there are some that are
> above perfectness, that they have more factors that they exceed 100%,
> whereas perfect
> numbers add up to 100%. And surprizingly this surplus and deficit
> holds true for odd numbers
> vying for perfectness. It seems strange that some odd numbers
> summation exceeds 100%.
>
> So I am cognizant of that fact, in marshalling this proof together.
>
> So basically let me summarize at this moment. There cannot be a odd
> perfect number
> except 1, because to attain perfectness the accounting must end up
> looking like this:
>
> 1/3 + 1/3combo + 1/3combo
>
> In the case of 45 we have:
>
> 1/45 + 3/45 + 15/45 + 5/45 + 9/45 for a total of 33/45
>
> We have a 1/3 in that of 15/45 and we have one 1/3combo in that of
> 1/45+5/45+9/45
> So in 45 we have 1/3 +1/3combo but no extra 1/3combo
>
> So perhaps I can shorten the proof by pointing out why no odd number
> can be perfect,
> except 1, because no odd number can add up to 1/3 + 1/3combo +
> 1/3combo. And also
> why this 1/3 has to be the unique adding up.
>
> I think the short answer is that you have to have 1/3 + 2/3combo and
> it is the 2 in the
> numerator that is never allowed to be a factor in odd numbers.
>
> So I suspect the entire proof of No Odd Perfect number hinges on that
> 2 in the numerator
> and why it is impossible for an Odd Perfect number except 1
>
>
>
>
> > (b) we have 1/2 + ( a combination equalling a sum of the other 1/2)
> >
> > Both those end up destroying the even versus odd factorability
> >
>
> Now the question would be, why AP able to prove this and noone before,
> since there
> was no mistakes in Logic Structure as in Euclid's IP of indirect
> method. Apparently a
> proof of No Odd Perfect Number just took a clever sort of fellow.
> Someone who can
> disassemble a math problem into its simple basic underpinnings.
>

Yes, the above constitutes a proof of No Odd Perfect Numbers. One last
detail I have to address is the uniqueness of the 1/3 +1/3combo
+1/3combo form.

It is not unique for there maybe a Odd Perfect Number candidate if we
jump to the
next series contender of 1/5 + 1/5combo + 1/5combo + 1/5combo +
1/5combo.
And the next contender would be a 1/7 series but they all fail
starting with the 1/3
series because they all require a even number in the numerator of
1/3 + 2/3combo or for 1/5 series we have 1/5 + 4/5combo

So that covers the uniqueness and covers all the contenders.

So in leaving, what is the impossibility of why No Odd Perfect Numbers
can exist
(other than 1) ? And the answer is that there is a fight, a tussle
between even and odd,
that in order to add up to 100% an odd number must be divisible by 2
and that is impossible.
In order for any odd number with its factors adding up to reach 100%
with no deficit and no
surplus but exactly 100% means there was a 2 in the numerator and thus
a 2 divides a odd
number.

P.S. really hard to fathom why these Perfect Numbers conjectures are
the oldest unproven
conjectures until now. I guess by superdeterminism, they were saved
for me to do. Oxygen
in me, oxygen of Plutonium, fill me with life anew, that I may love
what thou dost love and do
what thou has superdetermined to do.

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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