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From: Archimedes Plutonium on 8 Aug 2010 13:50
--- quoting parts of old post ---
From: Archimedes Plutonium <plutonium.archime...(a)gmail.com>
Newsgroups: sci.math,sci.logic,sci.edu
Subject: attempts by Eagle, Flath, and Hardy, #29; 2nd ed; Euclid's
Infinitude
of Primes Proof Corrected
Date: Sun, 30 Aug 2009 21:27:13 -0700 (PDT)
Organization: http://groups.google.com
Lines: 162

(snipped)
(# 9) --- quoting Daniel E. Flath INTRODUCTION TO NUMBER THEORY, 1989
page
2
---
Theorem 2.2 Euclid. There are infinitely many primes.
Proof. We shall show that every finite set of primes omits at least
one
prime. It will follow that no finite set can contain all the primes.
Let {p_1,p_2,...,p_r} be a finite set of prime numbers.
By Theorem 2.1, (Every positive integer n greater than 1 is a product
of prime number.) , there is a prime divisor q of N = p_1*p_2*..*p_r
+1. Because q divide into N but p_i does not divide into N, the prime
q
must be different from p_1,p_2,...,p_r.
--- end quoting INTRODUCTION TO NUMBER THEORY, Flath ---

Amazing, simply amazing that Flath by 1989 delivers a valid Euclid
Infinitude of Primes proof direct method.. So there are people who
have a clear mind of
logic to prove Euclid's IP. Flath indicates Euclid gave a Direct
Method
proof although he does not outright say so or analyze the history as
to why so many mistakenly think and believe Euclid made an indirect
method proof.

Flath could improve his above by simply stating that the direct method
is a set cardinality increase of any finite set which then
universalizes
into an infinite set. And also, Flath could vastly improve his above
if
he had given an Indirect Method proof to contrast his valid Direct
Method proof.

--- end quoting old post ---

I need to include the attempts by Iain Davidson as per Euclid
Infinitude of Primes proof because the pattern of
Davidson of using "every number has at least one prime factor" was
also widely misused. So I need to include that pattern of logic
mistake in this history survey.

I need to clear out alot of the posts of the previous edition over
this "mistake of prime factors".

In general, the mistake of Davidson is thinking that a Lemma-- every
number has at least one prime divisor. The mistake that Davidson makes
is that he omits the fact that every number is divisible by itself
must be included in the Lemma.

Here is how the Lemma truly stands: Lemma-- every
number is divisible by at least one prime divisor and every number is
divisible by itself.

So the use of that Lemma in the Indirect Method, does not achieve a
contradiction but only achieves the fact that Euclid's Number W+1 is a
necessarily new prime number. So Davidson wastes 7 steps in a attempt
to prove IP, only to come to a conclusion that W+1 is necessarily
prime by using the Lemma.

Whereas I deliver a full proof in just 6 steps, which includes the
steps that W+1 is necessarily prime.

What Davidson offers in his many impolite and unmannered posts of
Euclid IP, what he offers is that
most people in mathematics are not very logical of mind and that we
need to be ultra logical in order to have a correct and valid proof.

Davidson was under a misguided assumption that a proof in mathematics
has a "menu of contradictions" to
select from in the reductio ad absurdum proof. That he can pick and
choose what his contradiction is going to be for Euclid's Infinitude
of Primes proof. It turns out that Euclid's IP has only one unique
contradiction, and not a menu to select from.

Part of the problem of modern day mathematics is that a math
curriculum does not require Symbolic Logic to become a mathematician.
It is assumed in the education curriculum of mathematics that students
have enough logic in their minds to not require Symbolic Logic as a
prerequisite of graduating in mathematics.

But as the proof of Euclid's Infinitude of Primes shows us, that no-
one in the history of mathematics had a valid Euclid IP proof Indirect
until the year 1991. And the second person in math history to do a
valid Euclid
Infinitude of Primes proof was Karl Heuer in 1994.

Much of this confusion about doing a valid Euclid IP would have been
alleviated if all mathematicians were required to take Symbolic Logic
as a prerequisite to becoming a mathematician.

It is very sad when I have to encounter thousands of so called trained
mathematicians telling me that in the
Indirect method that 16 is not necessarily a new prime number if 3 and
5 are the only primes that exist. This shows that when people do not
take Symbolic Logic, that they are really not mathematicians.

And the reason that Infinitude of Twin Primes was not proven until
1991 when the valid Indirect of Regular Primes infinitude was shown,
is because of the same reason-- mathematicians never really understood
the
valid Indirect method of Euclid IP. If 3 and 5 are the only primes
existing then both 14 and 16 are necessarily twin primes.

The Infinitude of Twin Primes proof is one of the world's most easiest
proofs of all. So simple that
anyone who can do a valid Euclid IP of regular primes
can in the turn of an instant, prove Twin Primes infinitude.

The history of Twin Primes conjecture is a history of
not understanding the Symbolic Logic behind the proof by
contradiction.

Which brings up Chandler Davis, Hardy/Woodgold of
Mathematical Intelligencer article of Fall 2009 and the insistence by
Davis to not include a citation reference of my work on this subject
that predates Hardy/Woodgold. Davis is violating codes of conduct by
his insistence of not referencing electronic newsgroups of sci.math.
Davis is thus "lifting or stealing" intellectual
property by refusing to cite earlier work that is
"published electronically in sci.math".

But I want to include Davis as one of those who could not do a valid
Euclid IP indirect method according to
email exchanges between Chandler Davis and myself.

I challenged Davis to email me his own version of a valid Euclid IP
indirect. He refuses to do so. So I openly challenge Davis to post to
sci.math his version of Euclid's IP indirect method.

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
From: sttscitrans on 8 Aug 2010 14:15
On 8 Aug, 18:50, Archimedes Plutonium <plutonium.archime...(a)gmail.com>
wrote:
> --- quoting parts of old post ---
> From: Archimedes Plutonium <#29;

(Usual mathematical flatulence deleted)

Your inability to understand a simple statement

"Every natural >1 has at least one prime divisor"

let alone prove or disprove it
leads me to believe that you do not even understand
your own proof.

If "Every natural >1 has at least one prime divisor"
is false, give a counterexample

n = ..... is a natural > 1 and n has no prime divisors.

Can you fill in the dots with a suitable number ?
Do you know what a prime divisor is ?

I have asked this question several times and you are incapable of
giving an answer.

Simply admit you have no idea what you are talking about.
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