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From: Archimedes Plutonium on 16 Jul 2010 04:59


Archimedes Plutonium wrote:
> Alright, so I have a method, now all I need are conjectures waited to
> be conquered after
> I sharpen my sword.
>
> So let me recap how the method works. I use Indirect Euclid Infinitude
> of Primes which retrieves two new primes not on the list I started
> with. I weave into the Indirect method the
> Mathematical Induction in case the primes need identification as a
> Euclid Number. This method proved the Infinitude of Mersenne Primes
> and the Infinitude of Perfect Numbers.
> So I take a peek at any other open conjectures of prime infinity.
>
> --- quoting in parts from Wikipedia with notes below ---
>
> Many conjectures deal with the question whether an infinity of prime
> numbers subject to certain constraints exists. It is conjectured that
> there are infinitely many Fibonacci primes[24] and infinitely many
> Mersenne primes, but not Fermat primes.[25] It is not known whether or
> not there are an infinite number of prime Euclid numbers.
>
> --- end quoting ---
>
> Fibonacci sequence and primes:
>
> 0,1, 1, 2 , 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .
>
> 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, ....
>
>
> Fermat primes F= (2^2^n) +1
>
> 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, …
>
> Euclid numbers
>
> The first few Euclid numbers are 3, 7, 31, 211, 2311, 30031, 510511
>
> --- end notes from Wikipedia ---
>
> Alright, the Math Induction was able to make the suppose true for N
> case, able
> to show that the N+1 case was another Mersenne form number of (2^p) of
> the
> (2^p)-1.
>
> Can I do the same for Fermat primes of form (2^2^n) +1? In other
> words, if I suppose
> true for the case N that Euclid's Number without the adding of 1 or
> subtracting of 1
> is another form of (2^2^n), then, just like Mersenne primes I will
> have proved an
> infinitude of Fermat primes. Quite honestly I think I can show that
> via Math Induction
> with the suppose for case N that the Euclid Number is (2^2^n) that the
> case N+1 can be
> finagled to be another form of (2^2^n), the reason I say this is
> because we can reiterate
> the primes listed in the sequence as many times as we like and when we
> divide them into
> Euclid's Number they still leave a remainder. For example, if 2,3 are
> the only primes that exist
> and I set up Euclid's Number to be (2x3x2x3x2x3x2x3...) then either
> add 1 or subtract 1, the
> division by 2 and 3 will still leave a remainder of 1. So in the case
> of Fermat's primes, it looks as though the Math Induction delivers a
> Fermat's prime for the N+1 case. And thus Fermat's
> primes are infinite, contrary to what Wikipedia wrote.
>
> As for Prime-Euclid-Numbers they are infinite set by the trivial proof
> from Twin Primes being
> infinite. So the proof is trivial since Twin Primes are infinite.
>
> As for the Fibonacci primes, I do not see them fitting into a Math
> Induction template. So I am
> going to sleep on this one.
>

Take my words back, I remembered something recursive with the
rectangles
and that the Fibonacci numbers generated Pythagorean triples.

--- quoting from Wikipedia ---
Any four consecutive Fibonacci numbers Fn, Fn+1, Fn+2 and Fn+3 can
also be used to generate a Pythagorean triple in a different way:

--- end quoting ---

So it looks good for a Math Induction to identify the N+1 case as a
Fibonacci number
but careful in what the N case is supposed. Of course the overhanging
Indirect Euclid
IP will ensure the Fibonacci number is prime.

So we are beginning to see the huge power that this team gives for
clearing out all prime
conjectures whose question mark is, is the set finite or infinite?

Had anyone made a count as to how many conjectures of prime set
infinite were outstanding?
Was it in the hundreds, perhaps thousands of unsolved infinity of
primes conjectures.

But I wish most people would apply a little commonsense to these open
conjectures, such as the case of Fermat's primes of form (2^2^n) +1.
Since the Polignac Conjecture is proved true
we can reach a point where N +2k of the separation between two primes
is far greater than
the separation of (2^2^n).

Now if I have time I would like to discuss a mental picture of how to
picture infinity as what
the old math implies what infinity looks like since they are derelict
in defining a boundary
between finite and infinite at 10^500. We can all mentally picture
10^500. But I would like to spend time on a picture of infinity that
the Old Math was belaboring under and which would easily clear up the
idea of how big (2^2^n) is in relation to infinity. And the picture I
give of
Infinity as how Old Math portrayed infinity can point out the flaws of
that viewpoint, and why
the 10^500 system is so superior to Old Math.

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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