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From: Archimedes Plutonium on 14 Jul 2010 05:57
Alright, I found a proof template for proving a broad class of prime
sets whether they
are infinite or not. The Twin Primes proof started it off and then the
Polignac Conjecture
was proved using the same technique of Indirect where W+1 and W-1 are
necessarily prime.
It works because the method delivers two necessarily primes.

But the method needs to be broadened to capture infinitude proofs of
other prime forms
such as the Mersenne Primes of form (2^p)-1 as this set of primes {3,
7, 31, 127, . . .}

Using that Twin Primes proof format, I cannot inject (2^p)-1 as the
last and final prime
in the series of all primes and then expect Euclid's Number to be a
prime of that form,
unless, however I sneek into the proof that of Mathematical Induction.
If I can do that
then the same template of proving can prove hundreds of infinitude
conjectures of various
forms of primes.

But to tell you the truth, I have the sneeky suspicion that the last
number in the prime
series of the Indirect proof I suspect we can say the prime that is
formed by Euclid's
Number can be said to be of the same form as the last and largest
prime in the series.
That is a hunch and not backed by any supporting evidence. And that is
why I am here
with injecting Math Induction to make the proof solid.



Archimedes Plutonium wrote:
> Archimedes Plutonium wrote:
(snipped)


> > Proof of Infinitude of Perfect Numbers and Infinitude of Mersenne
> > Primes:
> > (1) definition of prime
> > (2) hypothetical assumption: suppose set of all primes is finite
> > and 2,3,5, 7, 11, . ., ((2^p) - 1) is the complete list of all the
> > primes with
> > ((2^p) - 1) the last and largest prime.
> > (3) Form Euclid's numbers of W+1 = (2x3x5x 7x 11x . .x (((2^p) - 1))
> > +1
> > and W -1 = (2x3x5x 7x 11x . .x (((2^p) - 1)) -1
> > (4) Both W+1 and W -1 are necessarily prime because when divided by
> > all the primes that exist into W+1 and W-1 they leave a remainder
> > and so they are necessarily prime from (1) and (2)
> > (5) Contradiction to (2) that W+1 and W-1 are larger primes than
> > ((2^p) - 1).
> > (6) And W+1 is a prime of form (2^p) + 1, and W -1 is a prime of form
> > (2^p) - 1)
> > Reason: you can place any form
> > of algebraic prime (x^p) for the last prime in the series so long as
> > it is -1 or +1 addition
> > (7) Mersenne primes are an infinite set, hence Perfect numbers are
> > infinite set.

The proof method of Indirect gets two new necessarily primes. Trouble
with the above
is no guarantee that W+1 and W-1 have the Mersenne prime form. So that
is why
I need to inject Math Induction.

Now for this Mersenne prime proof and Infinitude of Perfect Numbers
proof, I need only W-1

Now I have not had practice with Math Induction for a long time so am
a bit rusty on this.
I am trying to streamline the situation. Whether to do the Math
Induction inside the reductio
ad absurdum or whether to do it outside the reductio ad absurdum?

I have the initial cases of 3, 7, 31, 127. Now I suppose true for the
last and largest prime
in the Series S as 2,3,5, 7, 11, . ., ((2^p) - 1). But what exactly do
I suppose true? Do I suppose that Euclid's Number in the Indirect came
out as a Mersenne prime of form
(2^p) -1 and larger than the Mersenne prime used in the proof? So that
when I stick this
second Mersenne prime into the series of all existing primes that the
math induction show
true for case n+1, that we can say it is true since the new Euclid
Number produces
a new and larger prime? Frankly, I believe that is true but shaky over
it, and need more of a
zugswang.

Note that the "p" in the Mersenne prime is a prime within the series
primes.

I am in no rush to do this incorporation. It is important because it
promises to prove perhaps
hundreds if not thousands of conjectures about prime sets whether
infinite or not.

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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