Prev: Euclid had a flaw in his proof of Infinitude of Primes since he was not privy to Unique Prime Factorization #630 Correcting Math
Next: maybe Weil was wrong and a proof of Uniqe Prime Factorization #632 Correcting Math
From: Archimedes Plutonium on 4 Jul 2010 17:03


Archimedes Plutonium wrote:
> Sorry about this for this becomes like a CSI detective story mystery
> with alot of subplots and
> channels. It is a better "spy story" than any spy mystery novel. Its
> ending is finally coming
> to front and center stage. Its ending is not happy but sad, for not
> only did most of the mathematics community unable to deliver a valid
> proof of Infinitude of Primes but that
> Euclid himself could not deliver a valid proof since he lacked the
> Fundamental theorem
> of Arithmetic-- unique prime factorization.
>
> Here are the two proof of Infinitude of Primes in both methods of
> direct/constructive proof
> and of the indirect/contradiction method.
>
>
> And Euclid's IP, 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.
>
>
> So in words, the 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.
>
>
> 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
>
>
> 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)
>
> Now if we take Ore's example of Euclid's proof of IP as a faithful
> representation and **translation** of Euclid's ancient proof and we
> further take the words and evaluation of Weil on page 5 of his book
> "Number theory" as true, and no reason not to accept his evaluation.
> Then we come to the conclusion that unless you know the Unique
> Prime Factorization theorem (UPFAT) of numbers before doing the
> Infinitude
> of Primes proof, that it is required in critical parts of the proof no
> matter
> if the direct method or indirect method is used. Both methods require
> UPFAT.
>
> --- quoting from Number Theory and Its History, Oystein Ore, 1948,
> page 65 ---
> Euclid's proof runs as follows: let a, b, c, . . ., k be any family
> of
> prime numbers. Take their
> product P = ab x . . x k and add 1. The P+1 is either a prime or not
> a
> prime. If it is,
> we have added another prime to those given. If it is not, it must be
> divisible by some prime
> p. But p cannot be identical with any of the given prime numbers a,
> b, . . ., k because then it
> would divide P and also P+1; hence it would divide their difference,
> which is 1 and this is impossible. Therefore a new prime can always
> be
> found to any given (finite) set of primes.
> --- end quoting Ore ----
>
>
> quote of Weil's book "Number theory", 1984,
>  page 5: "Even in Euclid,
>  we fail to find a general statement about the uniqueness of the
>  factorization of an integer into primes; surely he may have been
> aware
>  of it, but all he has is a statement (Eucl.IX.14) about the l.c.m.
> of
>  any number of given primes. Finally, the proof for the existence of
>  infinitely many
>  primes (Eucl.IX.20).. "
>
> Now we should not be alarmed that Euclid failed to deliver a valid
> proof, or a nearly
> valid proof with just a minor flaw in it, because the key essential
> ingredient is forming
> the number P+1, and in my estimate the forming of that number means
> the person is
> on the right track.
>
> And we should not be surprized because modern day mathematicians made
> many mistakes
> such as not knowing the difference between IP direct or indirect, let
> alone invoking
> UPFAT.
>
> And we note also, that Hilbert and others spent considerable time in
> cleaning up the geometry
> proofs of Euclid where many holes and gaps in reasoning took place. We
> should not expect
> Euclid's book and works to have been 100% true without any flaws or
> gaps. Likewise, we should expect a gap in Euclid's Infinitude of
> Primes proof.
>
> Not only was Euclid's proof of Infinitude of Primes a direct or
> contructive proof method, but
> it omitted or was flawed without the use of Unique Prime
> Factorization. Weil points this out.
> And we can see that Euclid did not have UPFAT because he refers to a
> lemma of contradiction with Ore's talk of P and P+1 and division of 1
> by a prime.
>
> Now the question is, can we prove the Unique Prime Factorization
> theorem from the idea
> in Euclid's/Ore's lemma of contradiction with P and P+1 as a
> foundation? Is UPFAT equivalent
> to that section of the proof by Euclid and Ore where they apply the
> lemma on P and P+1?
>
> I think the answer is no. That the lemma used by Euclid and Ore are
> not sufficient to be
> turned into the Unique Prime Factorization theorem. I say this because
> before the lemma is
> reached in Ore's above proof, he says this "If it is not, it must be
> divisible by some prime p." That statement by Ore (Euclid?) requires
> UPFAT as justification
> for making the statement. So that UPFAT enters the picture before the
> lemma is applied.
>
> So I conclude this by saying, that in order to be able to do a valid
> Infinitude of Primes proof
> that one needs to have Unique Prime Factorization Theorem present
> before IP is started.
> And I think that Weil is correct that UPFAT may have been
> subconsciously known in Ancient
> Greek times by Euclid, it was not consciously known and thus not
> proven.
>
> Maybe the Arabian mathematicians centuries after Euclid proved UPFAT,
> for I doubt that it
> required until the 1800s for Gauss to have proved UPFAT. Perhaps
> Fibonacci proved UPFAT,
> or Tartaglia? Probably those Arabic mathematicians proved it, a few
> centuries after Euclid.
>

Well here is what Wikipedia has to say about UPFAT (Fundamental
theorem of
Arithmetic) and its history:

--- quoting from Wikipedia ---
The theorem was practically proved by Euclid (in book 7 of Euclid's
elements, propositions 30 and 32), but the first full and correct
proof is found in the Disquisitiones Arithmeticae by Carl Friedrich
Gauss. It may be important to note that Egyptians like Ahmes used
earlier practical aspects of the factoring, such as the lowest common
multiple, allowing a long tradition to develop, as formalized by
Euclid, and rigorously proven by Gauss.

--- end quoting ---

From the way that Euclid handled his Infinitude of Primes proof, is
evidence that UPFAT was
not known to ancient Greece for then it would have no need of the
lemma in Ore's/ Euclid's
proof.

And I doubt that mathematics history was without a proof of UPFAT
until Gauss in the 1800s
performed. I suspect someone gave a proof within a hundred or 200
years circa Euclid.

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
 | 
Pages: 1
Prev: Euclid had a flaw in his proof of Infinitude of Primes since he was not privy to Unique Prime Factorization #630 Correcting Math
Next: maybe Weil was wrong and a proof of Uniqe Prime Factorization #632 Correcting Math