Which is more flawed and erroneous, MathWorld's Euclid Infinitude of Primes or Wikipedia's
a_plu...@hotmail.com
--- quoting the current Wikipedia page 15MAR06 on Euclid's Infinitude
of Primes proof ---
How many prime numbers are there?
There are infinitely many prime numbers. The oldest known proof for
this statement 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.
--- end quoting ---
--- quoting MathWorld's Euclid Infinitude of Primes proof of 15MAR06
---
Given a finite sequence of consecutive primes 2, 3, 5, ..., p,
the number
N==2.3.5...p+1, (1)
known as the ith Euclid number when p==p_i is the ith prime, is either
a new prime or the product of primes. If N is a prime, then it must be
greater than the previous primes, since one plus the product of primes
must be greater than each prime composing the product. Now, if N is a
product of primes, then at least one of the primes must be greater than
p. This can be shown as follows.
If N is composite and has no prime factors greater than p, then one of
its factors (say F) must be one of the primes in the sequence, 2, 3, 5,
..., p. It therefore divides the product 2.3.5...p. However, since it
is a factor of N, it also divides N. But a number which divides two
numbers a and b<a also divides their difference a-b, so F must also
divide
N-(2.3.5...p)==(2.3.5...p+1)-(2.3.5...p)==1. (2)
However, in order to divide 1, F must be 1, which is contrary to the
assumption that it is a prime in the sequence 2, 3, 5, .... It
therefore follows that if N is composite, it has at least one factor
greater than p. Since N is either a prime greater than p or contains a
prime factor greater than p, a prime larger than the largest in the
finite sequence can always be found, so there are an infinite number of
primes.
--- end quoting MathWorld ---
I am going to try giving my most elegant and best flowing proofs of
Infinitude of Primes both Direct and Indirect.
Direct method of Infinitude of Primes proof: We show that if we can
increase the set cardinality by one more new prime to any finite set of
primes means the primes are infinite. Given any finite set of primes we
multiply the lot and add 1. Denote this number as W+1. Either W+1 is
prime itself or has a prime factor not on the list of that finite set.
Hence the proof.
Indirect method of Infinitude of Primes proof: Define what it means to
be prime as a number divisible only by 1 and itself. Reductio Ad
Absurdum step says "Suppose false that the set of primes is not
infinite but finite". Then p_f is the last and final prime and the set
of all the primes that exists is 2,3,5,7, ..., p_f. Multiply the lot
and add 1 where W+1 = (2x3x5x7x...x p_f) +1. W+1 is necessarily a new
prime, and contradiction. Hence the set of all primes is infinite.
Neither MathWorld's nor Wikipedia's rendition are valid. Both are
flawed because they mix the Direct with the Indirect into one proof.
The difference between the Direct and Indirect is that the new number
W+1 is necessarily prime in the indirect, but W+1 is either prime or
has a prime factor in direct. This difference is important because the
Indirect Method leads to a quick and easy proof of the Infinitude of
Twin Primes were we construct W+1 and W-1.
Whoever wrote the MathWorld version of Euclid's Infinitude of Primes
must have entered some sort of contest who could write the ugliest
version of this proof. Not only is MathWorld's version invalid because
it is a hodgepodge mix of both direct and indirect. But that much of
MathWorld's version is the stating of "irrelevancies" within the proof
structure. The person that wrote this MathWorld entry seems to have a
fetish for stating that a number is not 1 itself. I mean every other
paragraph the author seems to think that numbers have to be
distinguished as not being 1. And for the first time in the world
someone thinks that chasing after "difference a-b" is of any relevance
to the proof at hand. The author of MathWorld's version may as well
have included a paragraph that "lilac bushes bloom in springtime".
I get the sense that MathWorld held a drunk on alcohol party and who
could write the most pathetic and messy Euclid's Infinitude of Primes.
As to Wikipedia's version, it is much better than MathWorld's but it is
still flawed and invalid because it is a mixture of both direct and
indirect. Wikipedia could just delete its opening word "Suppose" and
say it is a direct proof that increases set cardinality of any finite
set of primes.
Wikipedia's is better than MathWorld's but that is not saying much
because both are flawed and invalid.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
Ioannis
news:1142443911....@z34g2000cwc.googlegroups.com...
[snip]
> Wikipedia's is better than MathWorld's but that is not saying much
> because both are flawed and invalid.
How many different threads will you start on the same silly subject?
*PLONK^^2*
> Archimedes Plutonium
--
Ioannis
Jan Burse
> <a_plu...@hotmail.com> wrote in message
>>Wikipedia's is better than MathWorld's but that is not saying much
>>because both are flawed and invalid.
> How many different threads will you start on the same silly subject?
> *PLONK^^2*
Yeah Wikipedia itself has a discussion mechanism and more.
http://en.wikipedia.org/wiki/Wikipedia:Articles_for_deletion
Bertie Reed
> I am going to try giving my most elegant and best flowing proofs of
> Infinitude of Primes both Direct and Indirect.
>
> Direct method of Infinitude of Primes proof: We show that if we can
> increase the set cardinality by one more new prime to any finite set of
> primes means the primes are infinite.
What?
<snip>
>
> Whoever wrote the MathWorld version of Euclid's Infinitude of Primes
> must have entered some sort of contest who could write the ugliest
> version of this proof.
Oh I see. Parody.
> I get the sense that MathWorld held a drunk on alcohol party and who
> could write the most pathetic and messy Euclid's Infinitude of Primes.
> Archimedes Plutonium
> www.iw.net/~a_plutonium
> whole entire Universe is just one big atom
> where dots of the electron-dot-cloud are galaxies
Pretty good.
a_plu...@hotmail.com
Yeah Wikipedia itself has a discussion mechanism and more.
http://en.wikipedia.org/wiki/Wikipedia:Articles_for_deletion
A.P. writes:
I figured out why the Wikipedia and MathWorld Euclid Infinitude of
Primes is so weird, strange and has the feel of someone drunk writing a
proof. Is because both pages have sent authors to try to verbatim give
what Euclid gave over 2 milleniums as a proof. That is why the
MathWorld has a fetish over whether a number is not equal to 1, and
that is why the Arthur Rubin writeup of IP in Wikipedia is so odd is
because he tried to assemble a proof keeping to the original.
But in both cases, do we want to keep to the history and sacrifice and
lose validity or do we want to give a proof which is valid and true.
Because Euclid mixed methods and his own taken word for word is not a
valid proof. Much like several of Euclid's theorems in geometry are
invalid because they had hidden assumptions for which modern math
corrected.
We certainly give Euclid the credit for proving primes are infinite
because the heart of the proof is the forming of "multiply the lot and
add 1". But the assemblage of the proof would not pass muster in our
modern day logical syllogisms of Symbolic Logic.
So I think it is recommended that we give a version of Euclid IP that
is valid and true and does not follow the historical version, since so
many believe it was reductio ad absurdum, yet I believe it was direct
method.
The valid proof of IP is these two:
Direct Method: to any finite set of primes of cardinality n can be
increased by one more by forming a new number W+1 where we multiply the
lot of primes add 1. Either W+1 is prime itself or that W+1 has a prime
factor not on the finite list. Since any finite set of primes can be
increased by one more means primes are infinite.
The Direct Method is the one Euclid was writing.
Indirect Method: Definition of Primes. Next step is the reductio ad
absurdum where we suppose primes are not infinite but instead are
finite. Then p_f is the last and final prime and 2,3,5,7, ..., p_f is
all the primes that exist. Construct W+1 = (2x3x5x...x p_f ) +1 Because
of the definition of prime in step one W+1 is necessarily a new prime.
Contradiction. Hence primes are infinite.
I think it is appropriate for MathWorld and Wikipedia to give both the
direct and indirect method, for it allays so much of the confusion.
James McLaughlin
Infinitude of Primes section to be made unalterable to stop AP
tampering with it. Does anyone except AP have any reason why this
would not be a good idea, and is it possible for Wiki to "lock"
sections of a page without locking the rest of the page?
a_plu...@hotmail.com
A.P. writes:
It was lost to history as to the extent of the hatred towards people of
genius. We can never know how much hatred Democritus endured. Nor can
we fully appreciate the hatred endured by Galileo or even Newton. But
the world, for the first time, has a written electronic history of the
hatred that A. P. endured.
If any college professor of mathematics had done what I have done for
the Euclid Infinitude of Primes, they would have appeared on the front
cover of every science magazine and newspaper in the world. Instead,
when A.P. does something of huge importance, hatred and suppression
emerges. The reason for this is that I am the author of the Atom
Totality theory, and that these little-minded hatemongers cannot be
happy with the credit of anything that comes from Archimedes Plutonium.