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| Archimedes Plutonium... |
 Posted: Sun Aug 23, 2009 6:18 pm |
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I am not just complaining for I actually attempted on
several occasions to correct Wikipedia, but if you
have any experience with interfacing with those editors, they are like
bulldogs even when wrong. So it is better to write it up in Usenet
sci.math where your work is not reverted instantly.
Let me describe the evolution of the Wikipedia entry
on Euclid's Infinitude of Primes Proof for it has changed frequently
but essentially always a failed
and invalid attempt. There was a Number theory book written by three
authors of Michigan or Michigan State University and one of those
authors
was Montgomery which was a standard used textbook at college and their
proof of Euclid IP was also flawed. This Montgomery version was the
one
used by Wikipedia where they changed a few words around and plastered
it up on Wikipedia. That version
was the Wikipedia version for a long time and it was
essentially a failed attempt of a Indirect Method.
Now the current Wikipedia entry of Euclid IP is a
Direct Method of augmenting any finite set. Trouble with it though is
that they reference Hardy and Hardy
is notorious for writing in his book A Mathematician's
Apology that Euclid's IP is a gem of a proof using
reductio ad absurdum as the ultimate gambit method
of offering up all of math to yield a contradiction.
So here we have Wikipedia editors and likely that Arthur Rubin wrote
the entry, since he guards that
entry so painstakingly. But anyway, here we have
a entry that is Direct Method that refers to Hardy who
speaks of a Indirect Method proof for IP in his Apology book.
So where is the poor student wanting to learn Euclid's IP in all of
this back and forth contradictory
shuffled up menagerie of a Euclid Infinitude of Primes. They read a
Direct method in Wikipedia which refers to a Indirect Method source of
Hardy
and in the summary below the proof by Wikipedia
they yakk about 30031 confuses new students.
Well, yes of course the students are confused when
the professors of mathematics are too confused to
ever straighten out the mess.
Noone in the history of mathematics has ever said--
to do a valid Infinitude of Primes a la Euclid, you have to do both
the Direct and the Indirect simultaneously. Because only in this
manner is the author not confused. And hence the student who is
trying to learn the proof of Euclid's Infinitude of Primes has a
chance of not being confused.
When we have G.H. Hardy and Arthur Rubin and Bill
Dubuque unable or unwilling to give both Direct and
Indirect as a Euclid Infinitude of Primes, then of course, only
confusion abounds.
Archimedes Plutonium wrote:
[quote:6bddd3868b]Here is Wikipedia's currently flawed IP proof:
--- quoting Wikipedia Euclid IP on 22AUG09 ---
There are infinitely many prime numbers. The oldest known proof for
this statement, sometimes referred to as Euclid's theorem, is due to
the Greek mathematician Euclid. Euclid states the result as "there are
more than any given [finite] number of primes", and his proof is
essentially the following:
Consider any finite set of primes. Multiply all of them together and
add 1 (see Euclid number). The resulting number is not divisible by
any of the primes in the finite set we considered, because dividing by
any of these would give a remainder of 1. Because all non-prime
numbers can be decomposed into a product of underlying primes, then
either this resultant number is prime itself, or there is a prime
number or prime numbers which the resultant number could be decomposed
into but are not in the original finite set of primes. Either way,
there is at least one more prime that was not in the finite set we
started with. This argument applies no matter what finite set we began
with. So there are more primes than any given finite number. (Euclid,
Elements: Book IX, Proposition 20)
This previous argument explains why the product P of finitely many
primes plus 1 must be divisible by some prime (possibly itself) not
among those finitely many primes.
The proof is sometimes phrased in a way that falsely leads some
readers to think that P + 1 must itself be prime, and think that
Euclid's proof says the prime product plus 1 is always prime. This
confusion arises when the proof is presented as a proof by
contradiction and P is assumed to be the product of the members of a
finite set containing all primes. Then it is asserted that if P + 1 is
not divisible by any members of that set, then it is not divisible by
any primes and "is therefore itself prime" (quoting G. H. Hardy). This
sometimes leads readers to conclude mistakenly that if P is the
product of the first n primes then P + 1 is prime. That conclusion
relies on a hypothesis later proved false, and so cannot be considered
proved. The smallest counterexample with composite P + 1 is
(2 × 3 × 5 × 7 × 11 × 13) + 1 = 30,031 = 59 × 509 (both primes).
--- end quoting Wikipedia's flawed Euclid IP ---
[/quote:6bddd3868b]
Euclid's Infinitude of Primes Proof:
Direct method---
Given any finite set of primes we can augment
that set with one more new prime and thus since
we augment *any finite set* means it is infinite. Given a
set of finite primes we
multiply the lot and add 1, call it W+1. Either
W+1 is prime or W+1 has a prime factor not
in the original finite set because W+1 is not
divisible by any of those finite listed primes.
Indirect method---
Suppose set of all primes is finite. Multiply the lot
and add 1, call it W+1. This new number W+1 is necessarily prime
because of the definition-of-prime.
Contradiction. Hence set of
all primes is infinite.
So if Wikipedia put the above two proofs together as
the entry for Euclid's IP, then they would have a crystal clear, valid
proof.
But as it currently stands, Wikipedia does not have
a valid proof but a tangled up mess of both methods.
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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