analysis of why invalid Re: new standard for Euclid IP proof-- able to give both Direct and Indirect alongside one another Re: Correcting Wikipedia's Euclid's Infinitude of Primes Proof


Mark Nudelman wrote:
On 4/11/2007 12:06 AM, a_plutonium wrote:
(snipped)
Well it is too bad you interpreted that way. I can be awfully gruff
many a
times, because I lose patience and want to move on to other subjects.
But I never intend on being jerkish. I focus on science not on people
stuff.

I was becoming impatient with your continued estimate that the
Wikipedia
Euclid IP was sound and valid. And I was not about to be devoting some
ten or more posts on various details. So I asked for you to post your
own
very own version of what a valid IP in Direct alongside a Indirect. I
was
hoping that if you did such, that you would immediately see that the
Wikipedia offering was indeed a mishmash that was flawed and invalid.

Rather than spend 10 or 20 posts downstream in back and forth arguing.

Besides, the first reply to this thread from Joergenson of Denmark, I
answered
with saying that write out his Indirect alongside a Direct.

So this is the answer to this confusion, since Hardy says IP is
Indirect
and Stillwell and Mark Nudelman says it is Direct, that if one simple
gives
both versions alongside one another, it does not matter what Euclid
method
was, so long as we have two methods where both proofs are valid.

And that is the trouble with Wikipedia in that they give one version
that is
flawed and invalid.

I still don't think Wikipedia's proof is invalid. But here are my versions:


By contradiction:

Assume there are a finite number of primes.
Let N be the number of primes.
Let S be the set of all primes. S = {p_1, p_2, ..., p_N}.
Let g = (p_1 * p_2 * ... * p_N)+1.
Since g > p_i for all i (1 <= i <= N), g is not equal to any p_i and g
is not in S.
Since S was assumed to be the set of all primes, g must be composite.
Let k be the largest prime factor of g. For all i (1 <= i <= N), no p_i
is a factor of g, so k is not equal to any p_i and k is not in S.
But S was assumed to be the set of all primes, so k is not prime, which
contradicts k being g's largest prime factor.
Therefore the assumption is impossible, and there are not a finite
number of primes.
QED


Direct:

Let S be any finite set of N primes. S = {p_1, p_2, ..., p_N}.
Let g = (p_1 * p_2 * ... * p_N)+1.
g is either prime or composite.
Case 1. g is prime: Since g > p_i for all i (1 <= i <= N), then g is not
equal to any p_i and g is not in S.
Case 2. g is composite: Let k be the largest prime factor of g. For all
i (1 <= i <= N), no p_i is a factor of g, so k is not equal to any p_i
and k is not in S.
Thus in either case, there is a prime which is not S.
Therefore no finite set of primes contains all primes.
QED.

--Mark

Well it is good that Mark posted his two proofs of Euclid IP, one
Direct, the other Indirect. Even though
Mark may feel it a disaster, for it teaches us all something about
this vaunted proof.

For me, it teaches me how to instantly detect whether the two
arguments are valid, or whether they have
a flaw so bad that they are invalid. The detecting mechanism is
whether the person uses a Prime Factor
Search to fetch a new prime in both methods. And that is what Mark had
to resort to doing which renders
his proof invalid.

To obtain two valid proofs the Prime Factor Search occurs only in the
Direct Method. In the Indirect Method,
the new prime is got by the fact that the reductio ad absurdum
assumption step guarantees the number
W+1 (Mark calls it g) is necessarily prime.

Both methods use W+1 which means both methods are logically linked.
Which means that to fetch the new
prime can not be got by the same Prime Factor Search. So when anyone
proffers up to proofs, one Direct
and one Indirect, they both have W+1, which inexorably forces the
mechanism to obtain a new prime not on
the list by a Prime Factor Search in the Direct and that W+1 is
necessarily prime in the Indirect.

If you have a Prime Factor Search in both, then you messed up, because
then there is no difference between
Direct and Indirect.

So Mark has done a valuable service by offering up his versions. And
this is how all future profferings of Euclid's
Infinitude of Primes Proof should be given--- Direct alongside
Indirect so that the author will end up with a crystal
clear and valid proof argument.

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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