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From: Archimedes Plutonium on 16 Jul 2010 04:14
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.

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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