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

On 4/9/2007 11:25 PM, a_plutonium wrote:
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.

The only problem with the wikipedia proof is a slight ambiguity in
language. The first sentence "Suppose you have a finite number of
primes" simply means "take a finite number of primes". It does not mean
"suppose the set of all primes is finite". If you accept the first
interpretation, then the proof is fine. It proves that given any set of
primes, there is another prime not in the set. Therefore, the set of
all primes cannot be finite. This is indeed the way Euclid did it --
see http://farside.ph.utexas.edu/euclid/euclid.pdf proposition 9.20.
Actually Euclid's proof is incomplete because he only proved that given
any set of exactly three primes, there is another prime not in the set.
The proof works only if the set contains any number of primes, but
Euclid apparently assumed the reader would extrapolate from his proof
with three primes. This is a well known failing of Euclid's proof. The
wikipedia proof is an improved version of Euclid's.

--Mark
.

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