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From: Archimedes Plutonium on 11 Jul 2010 23:16


Archimedes Plutonium wrote:
> Due to a conversation with a journal editor, my interest has been
> rekindled in a proof of the Infinitude of Twin Primes and all even
> numbered pairs of primes. The reason this attempt works, I feel, is
> that I have eliminated the regular primes out of the picture by a
> clever devise that was used in the Direct method of the
> square root to eliminate factors.
>
> XXXXXX
> Euclid's Infinitude of Primes proof, Direct or constructive in short-
> form goes like this:
>   1) Definition of prime
>   2) Given any finite set of primes
>   3) Multiply the lot and add 1 (Euclid's number) which I call W+1
>   4) Either W+1 is prime or we conduct a prime factor search
>   5) this new prime increases the set cardinality by one more prime
>   6) since this operation of increasing set cardinality occurs for
> any
>   given finite set we start with, means the primes are infinite set.
>
> XXXXXX
>
> Euclid Infinitude of Primes proof, Indirect in
>  short-
>   form goes like this:
>
>
> 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
>   5) contradiction to P_k as the last and largest prime
>   6) set of primes is infinite.
>
> XXXXXX
>
> DIRECT Method (constructive method), long-form; Infinitude of Primes
>  Proof
>
>
> (1) Definition of prime as a positive integer divisible
>   only by itself and 1.
>
>
> (2) Statement: Given any finite collection of primes
>  2,3,5,7,11, ..,p_n possessing a cardinality n Reason: given
>
>
> (3) Statement: we find another prime by considering W+1 =(2x3x...xpn)
>   +1 Reason: can always operate on given numbers
>
>
> (4) Statement: Either W+1 itself is a prime Reason: Unique Prime
>  Factorization theorem
>
>
> (5) Statement: Or else it has a prime factor not equal to any of the
>   2,3,...,pn
>  Reason: Unique Prime Factorization theorem
>
>
> (6) Statement: If W+1 is not prime, we find that prime factor Reason:
>  We take the square root of W+1 and we do a prime search through all
>  the primes from 2 to
>  square-root of W+1 until we find that prime factor which
>  evenly divides W+1
>
>
> (7) Statement: Thus the cardinality of every finite set can be
>  increased. Reason: from steps (3) through (6)
>
>
> (8) Statement: Since all/any finite cardinality set can be increased
>  by one more prime, therefore the set of primes is an infinite set.
>  Reason: going from the existential logical quantifier to the
>  universal
>  quantification
>
> XXXXXX
>
> INDIRECT (contradiction) Method, Long-form; Infinitude of Primes
> Proof
>  and
>  the numbering is different to show the reductio ad absurdum
> structure
>  as
>  given by Thomason and Fitch in Symbolic Logic book.
>
>
> (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
>
>
> (3.0) Suppose finite, then 2,3,5, ..,p_n is the complete series set
>   with p_n the largest prime Reason: this is the supposition step
>
>
> (3.1) Set S are the only primes that exist Reason: from step (3.0)
>
>
> (3.2) Form W+1 = (2x3x5x, ..,xpn) + 1. Reason: can always operate and
>   form a new number
>
>
> (3.3) Divide W+1 successively by each prime of
>   2,3,5,7,11,..pn and they all leave a remainder of 1.
>   Reason: unique prime factorization theorem
>
>
> (3.4) W+1 is necessarily prime. Reason: definition of prime, step
>  (1).
>
>
> (3.5) Contradiction Reason: pn was supposed the largest prime yet we
>   constructed a new prime, W+1, larger than pn
>
>
> (3.6) Reverse supposition step. Reason (3.5) coupled with (3.0)
>
>
> (4) Set of primes are infinite Reason: steps (1) through (3.6)
>
> XXXXXX
>
>
> For years now I thought I had not delivered a proof of the Infinitude
> of Twin Primes, that somehow I came up
> short, but due to a email conversation, I realized that
> all along I had proven the Infinitude of Twin, Quad, 6th primes
> and all other even multiples Primes.
>
> The proof is only Indirect method because only in the Indirect are you
> ensured of two new primes.
>
> Let me show you the Indirect Regular Primes Infinitude proof with a
> number example:
>
>
> 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.
>
> That number example is what delivers a valid Infinitude of Twin
> Primes, Quad Primes, 6th Primes, etc etc.
>
> XXXXXX
>
> 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
>
>
> (4.1) Set S are the only primes that exist 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
>

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

Bill, would you evaluate whether I cleared that up?


>
> (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
>
> (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 twinprimes 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
>
> Now, however there maybe a sticking point as to the application of the
> square root so as to keep higher primes of regular primes from
> interfering into the proof.
>
> I will probably have to make piecemeal corrections in the above so do
> not let the above be the final word.
>

Bill, I am hoping you can render a opinion as to whether the square
root patch
above elminates the pesky regular primes out of the picture?

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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