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From: Archimedes Plutonium on 14 Jul 2010 01:11


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.

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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Prev: Math-induction to get 2^p -1 Re: Proof of the Infinitude of Perfect Numbers and infinitude of Mersenne primes #668 Correcting Math
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