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From: Archimedes Plutonium on 27 Jun 2010 03:40
Now I have been posting to sci.math over the errors of Euclid's
Infinitude of Primes proof
(IP) since about 1993 to present, and wrote an entire book on this,
and pointed out the
mistakes that math professors were making on Euclid's IP proof. Now I
had every right
to do this because I gave both methods of Euclid's IP proof. But in a
recent article
to Mathematical Intelligencer (MI) by Michael Hardy and Catherine
Moongold, they do not provide both proof methods, nor do they even
provide their own but refer to others such as
Ore. They lambast many math professors. Now I feel they have no right
to do that because
they never provided their own renditions of both methods. I had the
right to lambast because
I furnished valid proofs both constructive and by contradiction (I
called them direct and indirect).

From reading the MI article, I doubt that Hardy and Moongold could
even do a valid indirect method, since in their article they say that
Euclid's number is not necessarily prime. That tells
me they do not know the valid proof of Euclid's IP indirect. And
obviously the editor of MI
does not know a valid IP indirect proof, or he would have stopped the
publishing of the article.

So I think that the editors of MI, and Hardy and Moongold, have no
right in lambasting their long list of mathematicians who they claim
made errors. I agree with this article that Euclid's IP was a
constructive (direct method) proof.

And I feel that this article is a lifting of my work on this subject
from my posts to sci.math
from 1994 to present. I am politely saying "lifting" but others can
call it a stealing of my work.
But then again, Hardy and Moongold never provide their own renditions
of the proof, either
constructive or contradiction.

And as I remarked from the fact that they say Euclid's number P + 1 is
not necessarily prime
indicates that Hardy, Moongold and the Mathematical Intelligencer
editors are too inept at
producing a valid Euclid IP by contradiction.

Here is my versions of both methods, in long form and short form.


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: numbers are either
unit, composite or prime

(5) Statement: Or else it has a prime factor not equal to any of the
2,3,...,pn
Reason: numbers are either unit, composite or prime

(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

(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: can always operate

(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)



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.

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.

I am still writing MI a letter, telling them to respectfully add a
correction in
a upcoming issue where the Correction lists Archimedes Plutonium as a
reference
to my sci.math postings over this work on Euclid IP proofs for that
Hardy/Moongold article. I have intellectual
property rights of the ideas I posted on Euclid's IP proof and my
ideas that Euclid
was a constructive proof and why mathematicians got it wrong was
posted many
years earlier than ever did Michael Hardy and Catherine Moongold and
the editors
of MI write their article.


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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