Correcting Wikipedia's Euclid's Infinitude of Primes Proof; David Eppstein (Wikipedia editor) is utterly wrong about Infinitude of Primes

Here is what Wikipedia editor David Eppstein has for this proof
---- quoting current Wikipedia ---



06:05, 10 April 2007 David Eppstein (Talk | contribs) m (revert
psychoceramic)

[edit] There are infinitely many prime numbers

The oldest known proof for the statement that there are infinitely
many prime numbers is given by the Greek mathematician Euclid in his
Elements (Book IX, Proposition 20). Euclid states the result as "there
are more than any given [finite] number of primes", and his proof is
essentially the following:

Suppose you have a finite number of primes. Call this number m.
Multiply all m primes together and add one (see Euclid number). The
resulting number is not divisible by any of the finite set of primes,
because dividing by any of these would give a remainder of one. And
one is not divisible by any primes. Therefore it must either be prime
itself, or be divisible by some other prime that was not included in
the finite set. Either way, there must be at least m + 1 primes. But
this argument applies no matter what m is; it applies to m + 1, too.
So there are more primes than any given finite number.

This previous argument explains why the product of m primes plus 1
must be divisible by some prime not among the m primes, or be prime
itself. A common mistake is thinking Euclid's proof says the prime
product plus 1 is always prime.
(2 · 3 · 5 · 7 · 11 · 13) + 1 = 30,031 = 59 · 509 (both primes) shows
this is not the case.


--- End quoting current Wikipedia ----

Really sad when people like David Eppstein reverts what is true, into
his utterly false mathematics. A case in
which someone who does not know what they are talking about, and who
has the gall to call me
"pyschoceramic". So this is telling of the calibre of Wikipedia
editors. That they retain editors who are
ill suited to understand this proof and then call others a bad name.

In the above IP, is a mix mesh of two methods in one. There is the
Direct in a prime factor search and then
David Eppstein starts the proof as a Indirect with "Suppose".
Apparently noone taught David that you cannot
have a combination of methods all in one proof.


Below is what I contributed to Wikipedia which is the world's first
clarification of Euclid's Infinitude of Primes
Proof. It shows both methods and it shows why you cannot mix the two
as one.

I am not going to fight with editors of Wikipedia who really have no
expertise in a subject that is foreign to
them and can revert the truth of a topic. But this needs to be done
because we cannot forever be teaching
young people the old mistaken ways forever.

--- quoting my Wikipedia correction ---

[edit] The number of prime numbers

[edit] There are infinitely many prime numbers

The oldest known proof for the statement that there are infinitely
many prime numbers is given by the Greek mathematician Euclid in his
Elements (Book IX, Proposition 20). Euclid states the result as "there
are more than any given [finite] number of primes".

Here mathematics runs into trouble, because the Euclid Infinitude of
Primes Proof has been misunderstood up until the 1990s. Most
mathematicians thought the Euclid proof was Reductio Ad Absurdum or
Indirect Method. But when one analyzes the language and thought
pattern delivered by Euclid, one realizes it is a Direct Proof Method.
So a valid Euclid Infinitude of Primes proof was not made clear until
Archimedes Plutonium debated it on the Internet in 1990s and wrote a
book on this proof as such:

Monograph-Book: "Correcting the Logical Flaws of Euclid's Infinitude
of Primes Proof", Archimedes Plutonium Internet book published
1993-2006 (assimilated in 2006 in sci.math)

We are going to give Euclid credit for proving the Infinitude of
Primes, and we are going to toss out the wrangling argument as to
whether Euclid meant a Direct Proof or Indirect Proof and simply say
that Euclid got enough of the basic proving elements that Euclid
understood his proof to be true.

According to the book authored by Archimedes Plutonium here are the
two valid proof methods of Euclid Infinitude of Primes:


Here is the valid Direct proof of IP Infinitude of Primes Proof,
DIRECT Method

(1) Statement: Given any finite collection of primes 2,3,...,pn
possessing a cardinality n Reason: given

(2) Statement: we find another prime by considering W+1 =(2x3x...xpn)
+1 Reason: can always operate on given numbers

(3) Statement: Either W+1 itself is a prime Reason: numbers are either
unit, composite or prime

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

(5) Statement: If W+1 is not prime, we find that prime factor Reason:
Unique Prime Factorization theorem

(6) Statement: Thus the cardinality of every finite set can be
increased. Reason: from steps (3) through (5)

(7) Statement: Since all/any finite cardinality set can be increased
by 1 more therefore the set of primes is an infinite set. Reason:
going from the existential logical quantifier to the universal
quantification


Here is the valid Indirect proof of IP. (Using Hardy's terminology)
Infinitude of Primes Proof, INDIRECT Method

Anyway here is a streamlined proof of IP indirect method:

(1) The prime numbers are the numbers 2,3,...,pn,... of set S Reason:
definition of primes

(2) Suppose finite, then 2,3,...,pn is the complete series set Reason:
supposition step

(3) Set S are the only primes that exist Reason: from step (2)

Note: This step follows immediately from the "Suppose primes are
finite". And it is a very important step and because it is so much
like the statement "Suppose the set S of primes is finite", for it is
this statement "The set S is all the primes that exist". And it is
this step that disallows any other prime factor search once W+1 is
formed. W+1 is your only allowable prime candidate in the indirect
method, otherwise you are self-contradicting your own logic. The math
logic is supposed to get you the contradiction, not you, in your own
illogic.)

(4) Form W+1 = (2x3x,...,xpn) + 1 Reason: can always operate and form
a new number

(5) Is W+1 prime? yes Reason: Unique Prime Factorization theorem
combined with the definition of what prime is.

(6) Contradiction Reason: pn was supposed the largest prime yet we
constructed a new prime larger than pn

(7) Reverse supposition step and primes are infinite Reason: steps (1)
through (6)

Even MathWorld and Wikipedia have failed to give a valid proof of
Infinitude of Primes where they mixed the two methods and bunched them
together as one. Unable to distinguish what Direct and Indirect
methods are.

The reason the mathematics community continues to fail in delivering a
valid proof of the Infinitude of Primes is because they seem to never
understand that W+1 is always a new prime in the Indirect Method. And
their illogic keeps them harping back to a mistake ridden case
example:

Quoting the last author in Wikipedia who does not understand a valid
proof of Infinitude of Primes: "(2 · 3 · 5 · 7 · 11 · 13) + 1 = 30,031
= 59 · 509 (both primes) shows this is not the case"

He does not understand that in the Indirect Method, that 30,031 is
necessarily a new prime.

In the Direct Method proof there is a prime factor search and that
30,031 is not necessarily prime and the proof method resolves this
issue. But in the Indirect Method all of the W+1 are new primes, no
matter what their actual number is, such as say 30,031, and the method
is what makes the proof because of the reductio ad absurdum
assumption. So there never is a prime factor search in the Indirect
Method because the moment W+1 is formed, it is a new prime not on the
original list.

Other mathematicians have given their own proofs. One of those (due to
Euler) shows that the sum of the reciprocals of all prime numbers
diverges to infinity. Kummer's is particularly elegant and Harry
Furstenberg provides one using general topology.[2][3]

--- end quoting ---

Archimedes Plutonium
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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