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From: Archimedes Plutonium on 12 Jul 2010 02:49
Alright I gave this as an example of how Regular primes infinitude
proof works in Indirect:
> > Euclid Infinitude of Primes proof, Indirect in
> >  short- form with number example of 3 and 5 :
> >
> >
> > 1) Definition of prime
> >   2) Hypothetical assumption, suppose set of primes 3,5 are all the
> > primes that exist with 5 the largest prime
> >   3) Multiply the lot and add 1 (Euclid's number) which is (3x5) +1 =
> > 16
> >   4) 16 is necessarily prime due to (1) and the assumptive step
> >   5) contradiction to 5 as the last and largest prime
> >   6) set of primes is infinite.

Now let me try that with Infinitude of Twin Primes with the example of
2,3,5

(1) Definition of prime
(2) Hypothetical Assumption, suppose set of all Twin Primes is finite
with
3,5 being the last two twin primes of the sequence set S = 2, 3, 5
where
5 is the last and largest of the twin primes
(3) Multiply the lot and add 1 and subtract 1, yielding W+1 = 2x3x5
+1 = 31 and
W-1 = 2x3x5 -1 = 29
(4) Take the square root of W+1 and W-1, and there cannot be any
regular primes
for consideration of being a prime factor of W+1 or W-1
(5) Successively divide all the primes in the sequence S into W+1 and
W-1 and
they all leave a remainder of 1
(6) The two new numbers W+1 and W-1 are necessarily two new twin
primes
(7) Contradiction to 3 and 5 with 5 the largest and last twin prime
(8) Set of Twin Primes is infinite.

Yes, I am now confident that I finally have found a proof of the
Infinitude of Twin
Primes.

What kept me back for years was the nagging suspicion that the
surrounding "regular"
primes was interfering with the proof focused on Twin primes. So to
get rid of the
surrounding regular primes apply the square root test to the W+1 and
W-1





> > Proof of the Infinitude of Twin Primes:
> >
> > INDIRECT (contradiction) Method, Long-form; Infinitude of Twin Primes
> >
> >
> > (1) Definition of prime as a positive integer divisible
> >   only by itself and 1.
> >
> >
> > (2) The prime numbers are the numbers 2,3,5,7,11, ..,pn,... of set S
> >   Reason: definition of primes
> >
> > (3) Let us instead pick the numbers of primes as
> > the succession of 2,3,5,7,. . , p(n), p(n+2) where
> > the p(n) and p(n+2) are twin primes
> >
> >
> > (4.0) Suppose twin primes are finite, then 2,3,5, ..,p_n ,
> > p_n+2 is the complete series set
> >   with p_n and p_n+2 the last and largest twin primes Reason: this is
> > the supposition step
> >

I should say the hypothetical supposition step

> >
> > (4.1) Set S are the only primes that exist

here I should say Set S are the only primes that exist between 2 and
the last and
largest twin prime member p_n+2


Reason: from step (4.0)
> > This is the step in which I hesitated in calling
> > my proof a genuine proof because I pictured larger regular primes
> > beyond the p_n+2, but that was superfluous
> >

Yes, in years past, what frightened me was that the regular primes got
in
the way of saying that all the primes from 2 to p_n+2 were in that
sequence.


Why I did not make this statement in years past, I will never know,
for sometimes
it just takes an elapse of time for the mind to clear out and produce
the proof.

Set S are the only primes that exist between 2,3, . . p_n, p_n+2

> >
> > (4.2) Form W+1 = (2x3x5x, ..,xp_n x p_n+2) + 1.
> > And form W-1 = (2x3x5x, ..,xp_n x p_n+2) - 1.
> > Reason: can always operate and
> >   form a new number
> >
> >
> > (4.3) Divide W+1 and W-1 successively by each prime of
> >   2,3,5,7,11,..p_n+2 and they all leave a remainder of 1.
> >   Reason: unique prime factorization theorem
> >
> > Now here is where my previous proof attempts failed and here is the
> > patch I wish to apply to stop it from failing. If I apply a patch so
> > as to eliminate all the regular primes beyond p_n+2 then the proof
> > works.
> > And the way I do that is apply a square root to the
> > W+1 signifying that no primes above p_n+2 will be a factor of W+1 or
> > W-1
> >

Alright, this square root business is new to the Euclid Infinitude of
Primes
proof. Not new to me, but new to others because it is never mentioned
that
in Euclid's quest for a prime factor in the Direct method requires the
idea of
the Square Root of Euclid's number so as to check for the prime factor
missing
if it is missing. So in the case of 2x3x5 +1 = 31 the square root is
5.5... and thus
we are concerned only with numbers 5 or smaller as a prime factor.

And via this square root patch, I eliminate the regular primes that
could interfer with
my focus on twin primes.




> > (4.4) W+1 and W-1 are necessarily prime. Reason: definition of prime,
> > step
> >  (1).
> >
> >
> > (4.5) Contradiction Reason: p_n+2 was supposed the largest twin prime
> > yet we
> >   constructed a new twin primes, W+1 and W-1, larger than p_n+2
> >
> >
> > (4.6) Reverse supposition step. Reason (4.5) coupled with (4.0)
> >
> >
> > (5) Set of twin primes are infinite Reason: steps (1) through (4.6)
> >
> > XXXXXX
> >
> > Now a identical proof procedure works for Quad primes
> > of p_n and p_n+4, and for the 6th prime pairs of
> > p_n and p_n+6
> >

There maybe a sticking point about this proof procedure for large even
numbered
prime pairs such as say p_n and p_n+1000, that we may have to have
some
more qualifiers to impose the Square Root Patch, due to the large
spread between
the n and even numbered n_

Comments: Anyway from 1993 to this day, I have always believed in the
idea that
the tiny, however tiny difference of the proof of the Indirect Euclid
that W+1 was
necessarily prime in that method was the key to Twin Primes proof of
wrangling out
W -1 and W +1 to perform that proof. What dissuaded me for some years
was that
the regular primes may get the way or become distractive. But once a
way of eliminating
the regular primes, then the proof comes easily.

Ironically, just a few days ago I wrote words to the effect that Twin
Primes would never
have a proof due to imprecise defining of "finite versus infinite
number." If one plays
in any science long enough, the science makes fools out of all of us.

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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