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From: Archimedes Plutonium on 4 Jul 2010 16:39
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.


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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