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From: Archimedes Plutonium on 12 Jul 2010 18:35
Alright, I made a series of mistakes in the last several posts,
starting off by thinking that the
method that proves infinitude of Twin Primes is going to prove Quad
Primes, then N+6 primes
then etc etc.

When caught in confusion, it is best to walk away from it and let the
mind settle down, and
piece things together. Most thoughts of complex things are thoughts in
error, and only rarely
are thoughts about complex things correct.

I had a proof of the Infinitude of Twin Primes dating all the way back
to 1993. When the Euclid Infinitude of Regular Primes is done in the
Indirect Method, it actually is a proof, not of the Regular Primes,
for which it easily can be, but is a proof of the Infinitude of Twin
Primes. Because the very same mechanism that P+1 is necessarily prime
in that method, allows
for P-1 to be necessarily prime.

Euclid INDIRECT Infinitude of Regular Primes proof:
1) Definition of prime
2) Hypothetical assumption, suppose set of primes 2,3,5,7,.. is
finite with P_k the last and final prime
3) Multiply the lot and add 1 (Euclid's number) which I call W+1
4) W+1 is necessarily prime from the definition of prime and the
assumption space
5) contradiction to P_k as the last and largest prime
6) set of primes is infinite.

AP INDIRECT Infinitude of Twin Primes proof:
(1) Definition of prime
(2) Hypothetical assumption, suppose the set of primes 2,3,5,7,.. is
finite with P_n and P_n+2 as the last and final two primes in
existence
(3) Multiply the lot and add 1 and subtract 1 (Euclid's numbers) which
I will call W+1 and W-1
(4) W+1 and W-1 are necessarily two new primes from the definition of
prime
and from the assumption space
(5) Contradiction to P_n+2 being the last and largest prime
(6) Twin primes set is infinite

Now I do not know if in the whole of mathematics, of all its proofs
that the Infinitude of
Primes proof yields a asymmetry of result in that the Direct method
yields only a
infinitude of regular primes and cannot produce infinitude of twin
primes, whereas the
Indirect method can yield both.

So whether the Infinitude of Primes proof method of direct and
indirect is the only asymmetrical example or whether mathematics has
dozens of proofs where an asymmetry
between direct and indirect exists. I would guess asymmetry abounds.

Now my series of mistakes is all due to my stubborn wish that the
Indirect method proves
quad primes then N+6 primes etc etc.

We can clearly see why that is not the case. We never needed the
square root patch in indirect. What prevents a Quad Prime proof is
that W+2 and W-2 are no longer guaranteed to
be nondivisible by all the primes in the succession list S. We can no
longer say that when
dividing W+2 by all the primes that exist in S, leave a remainder. So
all we get as a proof
by Indirect Method is a Twin Prime proof.

But that is sufficient to prove the infinitude of Quad primes and all
primes of form N + 2k
for k>1. I say this because to have only Regular Primes infinite and N
+2 primes infinite
but to have N+4, N+6, etc etc as all finite sets of primes is a
contradiction. Whether a contradiction of the Prime distribution
theorem or some other distribution of primes theorem
I am not quite sure as yet.

What I suspect is that given the information that Twin Primes is an
infinite set, makes the
proposition that all N+2k, k>1 sets of primes must also be infinite.


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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