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From: Archimedes Plutonium on 15 Jul 2010 04:06
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.


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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