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From: Archimedes Plutonium on 22 Jul 2010 13:17


Archimedes Plutonium wrote:
(big snips)

>
> Here is the Indirect Method in short form:
>
> (1) definition of prime
> (2) hypothetical suppose all primes are finite with 2,3,5,7,.., p_k
> the complete list
> of primes with p_k the last and largest prime number
> (3) form Euclid's Number of multiply the lot and add 1 and call it W+1
> (4) W+1 is necessarily a new prime by (1) with (2)
> (5) contradiction to p_k being the last and largest prime since we
> have W+1
> (6) reverse the hypothetical supposition that primes are infinite.
>

Any mathematician, alive and working today, upon reflection, upon a
calm, relaxed
and cool reflecting back over the entire field of mathematics,
instinctly knows
that when a proof method is correct, and used to prove a statement,
that any corollaries of that
statement should also be proven by the same method.

Twin Primes infinitude is a subset or corollary statement of all the
Regular Primes infinitude. Euclid Direct method proof of infinitude of
Regular Primes is a valid proof method. But somewhere in that method
should and must lie a proof of Infinitude of Twin Primes. Or the
alternative, that the method of "multiply the lot and add 1" has a
flaw to it.

If we engineer a car that can go at the speed of sound, then as a
corollary, that car can
also go at the speed of 1/2 of the speed of sound.

Likewise, if Euclid's method can prove infinitude of Regular primes,
within that method
there must be a proof of Twin Primes since it is a corollary subset.

So where is the flaw in the method? Euclid's direct method has no
flaw, for it is constructive, but the flaw
lies in the Indirect. The flaw is that in the Indirect there is no
prime factor search. In the Indirect, W-1 and W+1 must be necessarily
two new primes not on any finite list of primes.

So the history of Twin Primes conjecture is a history of mostly not
recognizing a valid Indirect
method for Euclid's regular primes proof. It is a history that never
required a difficult proof. Never required an arsenal of new
mathematics. It only required a straightening out of a error filled
indirect method.

Good mathematicians, all of them should recognize that when you have a
proof of a statement in mathematics and you have a corollary
statement, regular-primes versus twin primes, that if
your method is valid, it should prove not only regular prime
infinitude but twin prime infinitude.

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
From: sttscitrans on 22 Jul 2010 15:14
On 22 July, 18:17, Archimedes Plutonium
<plutonium.archime...(a)gmail.com> wrote:

(Much drivel deleted)


In the Indirect, W-1 and W+1 must be necessarily
> two new primes not on any finite list of primes.

That, according to you, is the direct method.

You obviously do not understand the meaning of the word
ALL.

In one of your own examples, you assume that
3 and 5 are all the primes that exist.

PRIMES ={3,5}

Can you not follow that if 3 and 5 are the only
primes that exist, the other naturals are not primes.

NONPRIMES = {1,2,4,6,7,8,9,10,11,12, .....}

Perhaps think you that that numbers like 4 and 16
are prime nonprimes and should be in both sets -
PRIMES and NONPRINES ?

Have you worked out yet whether 1 is prime or not ?
You are very quiet on that topic.
Are you finding it too difficult to come up with
an adequate definition ?
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