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From: Archimedes Plutonium on 15 Jul 2010 05:18


Archimedes Plutonium wrote:
> Alright let see if I can start a rough draft of the proof of the
> Infinitude of Mersenne Primes
> and thus a proof of Infinitude of Perfect Numbers.
>
> I am calling it a template because I expect hundreds of proofs
> involving prime sets where the question is whether they are an
> infinite set. So I anticipate that these hundreds of conjectures
> will all flow through this same channel of proof where it is Indirect
> Euclid Infinitude of Primes
> format, coupled with a Mathematical Induction. I did not need Math
> Induction for Twin Primes
> nor for Polignac Conjecture, but when it came to Mersenne primes
> (2^p)-1, I was no longer
> sure that Euclid's Number fetched a Mersenne prime.
>
>
> Indirect Method Euclid Infinitude of Primes
> (1) definition of prime
> (2) hypothetical assumption step; suppose .. where last number in
> list
> is largest prime
> (3) form Euclid's Number/s
> (4) Euclid's Number/s are necessarily prime
> (5) contradiction to largest prime of list
> (6) set infinite
>
> Mathematical Induction procedure:
> (i) show true for initial cases of 1, 2, and even 3
> (ii) assume true for case of N
> (iii) must show true for case of N+1
>
> So let me just plaster, or shot up the rough draft of the proof that
> Mersenne Primes
> are infinite:
>
>
> (1) Definition of prime as a positive integer divisible
>   only by itself and 1.
>
>
> (2) The prime numbers are the numbers 2,3,5,7,11, ..,pn,... of set S
>   Reason: definition of primes
>
> (3) The Mersenne primes are of form (2^p) -1 and the first four are 3,
> 7, 31, 127
>
>
> (4) Suppose Mersenne Primes and regular primes are finite, then
> 2,3,5,7, ..,p_n is the complete series set
>   of Mersenne primes along with all the regular primes below p_n with
> p_n the largest Mersenne prime Reason: this is the supposition step
>
> (3.1) Set S are the only primes that exist Reason: from step (3.0)
>
>
> (3.2) Form W-1 = (2x3x5x, ..,xpn) - 1. Reason: can always operate and
>   form a new number
>
>
> (3.3) Divide W-1 successively by each prime of
>   2,3,5,7,11,..pn and they all leave a remainder of 1.
>   Reason: unique prime factorization theorem
>
>
> (3.4) W-1 is necessarily prime. Reason: definition of prime, step
>  (1).
>
> (3.5) Initial cases of Mathematical Induction
>
> The first few Mersenne primes are 3,7,31, 127
>
> So the initial case of a Math Induction works for Euclid's Number as
> W
> +1
>
> {2,3} are all the primes that exist yields Euclid Number (2x3)+1 = 7
> {2,3,5} are all the primes that exist yields Euclid Number (2x3x5)+1
> =
> 31
>
> (3.6) Assume true for case N of Mathematical Induction:
> assume true that the Euclid Number above of W-1 is of the form (2^p)-1
> and this further means that the Euclid Number of W-1 above means the
> series multiplication of (2x3x5x, ..,xpn) has the form of a number in
> the
> set (2^p) where p is prime.
>
> (3.7) Now must show true for Math Induction of N+1.
>
> (3.8) Include W-1 above into the new extended series set of
> {2,3,5,7, p_n, W-1} and translate into a new Euclid Number Y-1
> as this (2x3x5x, ..,xp_n x (W-1)) -1
>
> And now I am momentarily stuck and tired. I want to transition to
> where I say that
> the 2^p portion is squared in this new number of Y of the Y-1 and a
> second iterated Indirect
> method makes Y-1 also a new prime.
>
> Finally, infinitude of Mersenne primes.
>

Nice what a warm shower can do. The shower room seems to me to be a
thinking stimulant par excellence.

The above is new territory for me, although I vaguely remember going
through these
types of wanderings in early 1990s of interspersing Math Induction
into a proof.

Now some people may think that Math Induction is a indirect method
since they
see the words " suppose true for case N", but they are sadly mistaken,
for Math
Induction is but a definition of integer and is a rule-procedure.

And there is some spectacular bizarreness about Mathematical Induction
because
when it is true, means you have all the Natural Numbers and they are
infinite. So
one can ask, Mr. Plutonium, your use of Math Induction strives for
infinity and your
use of Indirect Euclid IP strives for infinity of Mersenne Primes. So
are you redundant
Mr. Plutonium?

And I answer no, I am not redundant in using Math Induction along with
Indirect Euclid IP.
I use Math Induction to insure that Euclid's Number is a Mersenne form
prime and I use
Indirect Euclid IP to guarantee that the Mersenne primes are infinite.

I am placing three Euclid IP proofs into the three steps of
Mathematical Induction.
There is no way that Math-Induction can tell me if the new number is
prime. And likewise,
there is no way that Indirect Euclid IP can tell me or insure that the
new Euclid Number
is of form (2^p)-1.

So it is a weaving together or braiding together of math induction
with a proof method of Indirect Euclid IP.

Now let me rewrite (3.8) above to read as follows:

(3.8) Include W-1 above into the new extended series set of
{2,3,5,7, p_n, W-1} and translate into a new Euclid Number Y-1
as this (2x3x5x, ..,xp_n x (W-1)) -1. And due to the Mathematical
Induction assume N true step that 2x3x5x, ..,xp_n is of form (2^p) of
a number
in this series 2,4,8,16,32,.... that the number W-1 is also
decomposable as
that of W = (2^p) so that we have ( 2^p)^2 (-1)

In step (3.8) I decompose the series into that of (2^p)(2^p) -1

Step (3.9) The square of a number in the series 2,4,8,16,32, ...
is also a member of that series

So finally in the step (4) the Mathematical Induction of show that p_N
+1 is
satisfied is true since Y-1 is that of the form (2^p)(2^p) -1

(4.1) Y-1 is necessarily a new prime number because all the primes
that exist
when divided into Y-1 leave a remainder
(4.2) Y-1 is a Mersenne prime because of Math-Induction steps
(4.3) Mersenne Primes are infinite because of the contradiction to the
supposition
that W-1 and then Y-1 were the last and largest Mersenne primes since
the Indirect
method reiterates another Mersenne Prime.
(5) Mersenne Primes are infinite
(6) Perfect Numbers are infinite

QED

Now I know the above has some sloppiness and gaps but the main ideas
are all there
and as the years go by I endeavor to cleanse and perfect the perfect
numbers proof.

And the learning curve is magnificent on this problem and proof
because we have
two things going on that are interwoven, we have Indirect Euclid IP to
insure a new prime
and we have Math Induction to insure a form for that new prime.

I do not think there is ever a proof in mathematics until now where we
have Math Induction
interwoven into a second method.


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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