Prev: Proof that 1 is the only Odd Perfect Number #665 Correcting Math
Next: improvement in Re: Proof of the Infinitude of Perfect Numbers and infinitude of Mersenne primes #667 Correcting Math
From: Archimedes Plutonium on 13 Jul 2010 16:40
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
(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

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
 | 
Pages: 1
Prev: Proof that 1 is the only Odd Perfect Number #665 Correcting Math
Next: improvement in Re: Proof of the Infinitude of Perfect Numbers and infinitude of Mersenne primes #667 Correcting Math